diff --git a/EuclideanGeometry/Check.lean b/EuclideanGeometry/Check.lean
index ab1ad822..2550a369 100644
--- a/EuclideanGeometry/Check.lean
+++ b/EuclideanGeometry/Check.lean
@@ -90,7 +90,7 @@ theorem let_test_prop (α : Prop) (a : α) (p : α → Prop) : (let x := a; p x)
     intro x
     exact x _
 
-def main_theorem (A B C : P) (h : ¬ collinear A B C) : Prop := by
+def main_theorem (A B C : P) (h : ¬ Collinear A B C) : Prop := by
   let hAB : B≠A := by exact (ne_of_not_collinear h).2.2
   let hCB : C ≠ B := by exact (ne_of_not_collinear h).1
   let l₁ := LIN A B hAB
@@ -99,7 +99,7 @@ def main_theorem (A B C : P) (h : ¬ collinear A B C) : Prop := by
   let E := Line.inx l₁ l₂ h'
   exact (E = B)
 
-example (A B C : P) (h : ¬ collinear A B C) : main_theorem A B C h := by
+example (A B C : P) (h : ¬ Collinear A B C) : main_theorem A B C h := by
   unfold main_theorem
   -- rw [let_test (α := B≠A) (a := (ne_of_not_collinear h).2.2) (p := fun x => let hCB := (_ : C ≠ B);
 -- let l₁ := LIN A B x;
diff --git a/EuclideanGeometry/Example/AOPS/Chap3.lean b/EuclideanGeometry/Example/AOPS/Chap3.lean
index 1d6f1fe0..9f909bed 100644
--- a/EuclideanGeometry/Example/AOPS/Chap3.lean
+++ b/EuclideanGeometry/Example/AOPS/Chap3.lean
@@ -15,7 +15,7 @@ Theorem: We have $BX = CY$ and $MX = MY$.
 -/
 
 -- Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
 -- Claim: $A \ne B$ and $A \neq C$. This is because vertices of nondegenerate triangles are distinct.
 lemma A_ne_B : A ≠ B := (ne_of_not_collinear hnd).2.2.symm
 lemma A_ne_C : A ≠ C := (ne_of_not_collinear hnd).2.1
diff --git a/EuclideanGeometry/Example/AOPS/Chap5.lean b/EuclideanGeometry/Example/AOPS/Chap5.lean
index d2fb19d3..75942170 100644
--- a/EuclideanGeometry/Example/AOPS/Chap5.lean
+++ b/EuclideanGeometry/Example/AOPS/Chap5.lean
@@ -12,7 +12,7 @@ namespace AOPS_Problem_5_7
 Prove that $\frac{EY}{EX}=\frac{AD}{DB} -/
 
 --Triangle A B C
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 lemma b_ne_a : B ≠ A := sorry
 lemma c_ne_a : C ≠ A := sorry
 lemma c_ne_b : C ≠ B := sorry
@@ -60,7 +60,7 @@ Prove that $IJ\prar BC$-/
 --It is simpler to use vectors but I think we should avoid vectors.
 
 --Nontrivial triangle A B C
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 lemma b_ne_a : B ≠ A := sorry
 lemma c_ne_a : C ≠ A := sorry
 lemma c_ne_b : C ≠ B := sorry
@@ -82,7 +82,7 @@ namespace AOPS_Problem_5_14
 Prove that AX^2 = BX \codt CX. -/
 
 -- In right triangle $\triangle PQR$, $\angle QPR = 90^{\circ}$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma a_ne_b : A ≠ B := sorry
 lemma b_ne_c : B ≠ C := sorry
@@ -102,7 +102,7 @@ $M$ is the midpoint of $AB$
 Prove that $MQ \parallel BC$ -/
 
 -- We have triangle $\triangle ABC$ with $AC \ne BC$
-variable {A B C : P} {hnd : ¬ collinear A B C} {hne : (SEG A C).length ≠ (SEG B C).length}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {hne : (SEG A C).length ≠ (SEG B C).length}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma a_ne_b : A ≠ B := sorry
 lemma b_ne_c : B ≠ C := sorry
diff --git a/EuclideanGeometry/Example/AOPS/Chap5a.lean b/EuclideanGeometry/Example/AOPS/Chap5a.lean
index 7d965d86..66c256df 100644
--- a/EuclideanGeometry/Example/AOPS/Chap5a.lean
+++ b/EuclideanGeometry/Example/AOPS/Chap5a.lean
@@ -18,7 +18,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Claim :$C \ne A$
   B_ne_A : B ≠ A :=
     -- This is because vertices $B, C$ of a nondegenerate triangle are distinct.
@@ -86,7 +86,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Claim :$C \ne A$
   B_ne_A : B ≠ A :=
     -- This is because vertices $B, C$ of a nondegenerate triangle are distinct.
diff --git a/EuclideanGeometry/Example/ArefWernick/Chap1.lean b/EuclideanGeometry/Example/ArefWernick/Chap1.lean
index 1d1f6f8b..94f7d839 100644
--- a/EuclideanGeometry/Example/ArefWernick/Chap1.lean
+++ b/EuclideanGeometry/Example/ArefWernick/Chap1.lean
@@ -14,7 +14,7 @@ In other words, let $\triangle A_1B_1C_1$ and $\triangle A_2B_2C_2$ be two trian
 Prove that $\triangle A_1B_1C_1$ is congruent to $\triangle A_2B_2C_2$. -/
 
 -- We have two triangles $\triangle A_1B_1C_1,A_2B_2C_2$ .
-variable {A₁ B₁ C₁ A₂ B₂ C₂ : P} {hnd₁ : ¬ collinear A₁ B₁ C₁} {hnd₂ : ¬ collinear A₂ B₂ C₂}
+variable {A₁ B₁ C₁ A₂ B₂ C₂ : P} {hnd₁ : ¬ Collinear A₁ B₁ C₁} {hnd₂ : ¬ Collinear A₂ B₂ C₂}
 -- $M_1,M_2$ are midpoints of $B_1C_1,B_2,C_2$
 variable {M₁ M₂ : P} {hm₁ : M₁ = (SEG B₁ C₁).midpoint} {hm₂ : M₂ = (SEG B₂ C₂).midpoint}
 -- We have $A_1B_1 = A_2B_2$, $A_1C_1 = A_2C_2$, and $A_1M_1 = A_2M_2$.
@@ -32,7 +32,7 @@ namespace Aref_Wernick_Exercise_1_2
 Prove that $\angle BFD = \pi / 2 - \angle CAB$. -/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $B \ne A$ and $C \ne A$ and $B \ne C$.
 --This is because vertices of nondegenerate triangles are distinct.
 lemma B_ne_a : B ≠ A := (ne_of_not_collinear hnd).2.2
@@ -60,8 +60,8 @@ lemma B_ne_f : B ≠ F := by
     have inxb : is_inx B (LIN B C (ne_of_not_collinear hnd).1) (LIN A B (ne_of_not_collinear hnd).2.2) := by
       exact ⟨(SegND B C (ne_of_not_collinear hnd).1).source_lies_on_toLine , (SegND A B (ne_of_not_collinear hnd).2.2).target_lies_on_toLine ⟩
     exact d_ne_b (unique_of_inx_of_line_of_not_para line_neq inxb.symm inxd.symm)
-  have bcd_notcoli : ¬ collinear B C D := (Line.lies_on_line_of_pt_pt_iff_collinear (b_ne_c (hnd := hnd)).symm D).mpr.mt d_not_lies_on_bc
-  have b_not_lies_on_cd : ¬ B LiesOn (LIN C D d_ne_c) := (Line.lies_on_line_of_pt_pt_iff_collinear d_ne_c B).mp.mt (flip_collinear_snd_trd.mt (flip_collinear_fst_snd.mt bcd_notcoli))
+  have bcd_notcoli : ¬ Collinear B C D := (Line.lies_on_line_of_pt_pt_iff_collinear (b_ne_c (hnd := hnd)).symm D).mpr.mt d_not_lies_on_bc
+  have b_not_lies_on_cd : ¬ B LiesOn (LIN C D d_ne_c) := (Line.lies_on_line_of_pt_pt_iff_collinear d_ne_c B).mp.mt (Collinear.perm₁₃₂.mt (Collinear.perm₂₁₃.mt bcd_notcoli))
   have f_lies_on_seg_cd : F LiesOn (SegND C D d_ne_c).1 := hf.2
   exact (ne_of_lieson_and_not_lieson (SegND.lies_on_toLine_of_lie_on f_lies_on_seg_cd) b_not_lies_on_cd).symm
 lemma d_ne_f : D ≠ F := sorry
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_1.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_1.lean
index 89a8d66e..b6e2c1c4 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_1.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_1.lean
@@ -24,8 +24,8 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   -- $A, E$ do not lie on $l$.
   A : Plane
   E : Plane
-  ABC_nd : ¬collinear A B C
-  EDF_nd : ¬collinear E D F
+  ABC_nd : ¬Collinear A B C
+  EDF_nd : ¬Collinear E D F
   -- need A and E be at the same side of l!!
   A_side : IsOnLeftSide A (SEG_nd B F)
   E_side : IsOnLeftSide E (SEG_nd B F)
@@ -34,11 +34,11 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   -- $BC = DE$
   h₂ : (SEG B C).length = (SEG D E).length
 attribute [instance] Setting1.B_ne_F
-lemma hnd₁ {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane}: ¬ collinear e.B e.A e.C := by
-  apply flip_collinear_fst_snd.mt
+lemma hnd₁ {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane}: ¬ Collinear e.B e.A e.C := by
+  apply Collinear.perm₂₁₃.mt
   exact e.ABC_nd
-lemma hnd₂ {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane}: ¬ collinear e.D e.F e.E := by
-  apply perm_collinear_trd_fst_snd.mt
+lemma hnd₂ {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane}: ¬ Collinear e.D e.F e.E := by
+  apply Collinear.perm₃₁₂.mt
   exact e.EDF_nd
 instance A_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane}: PtNe e.A e.B := ⟨(ne_of_not_collinear hnd₁).2.2⟩
 instance D_ne_E {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane}: PtNe e.D e.E := ⟨(ne_of_not_collinear hnd₂).2.1⟩
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_2.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_2.lean
index 55ab3ff4..ed7feca4 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_2.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_2.lean
@@ -20,8 +20,8 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   h₁ : (SEG A B).length = (SEG D C).length
   h₂ : (SEG D B).length = (SEG A C).length
   -- nondegenerate
-  hnd₁ : ¬ collinear D B A
-  hnd₂ : ¬ collinear A C D
+  hnd₁ : ¬ Collinear D B A
+  hnd₂ : ¬ Collinear A C D
   D_ne_A : D ≠ A :=(ne_of_not_collinear hnd₁).2.1
   -- $B,C$ is on the same side of line $AD$.
   B_side : IsOnRightSide B (SEG_nd A D D_ne_A)
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_3.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_3.lean
index 8ba41e4a..f0d4570b 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_3.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_3.lean
@@ -33,12 +33,12 @@ theorem Result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : ∠
   Because $DB = DC$ ,we have $\angle D B C = -\angle D C B$
   $\angle A B D = \angle A B C - \angle D B C = \angle D C B - \angle A C B = --\angle A C D$
   -/
-  have h₁ : ¬ collinear e.A e.B e.C := by
+  have h₁ : ¬ Collinear e.A e.B e.C := by
     exact (Quadrilateral_cvx.not_collinear₁₂₄ (QDR_cvx e.A e.B e.D e.C e.cvx_ABDC))
-  have h₂ : ¬ collinear e.D e.B e.C := by
-    have h₂' : ¬ collinear e.B e.D e.C := by
+  have h₂ : ¬ Collinear e.D e.B e.C := by
+    have h₂' : ¬ Collinear e.B e.D e.C := by
       exact (Quadrilateral_cvx.not_collinear₂₃₄ (QDR_cvx e.A e.B e.D e.C e.cvx_ABDC))
-    apply flip_collinear_fst_snd.mt h₂'
+    apply Collinear.perm₂₁₃.mt h₂'
   have C_ne_B : e.C ≠ e.B := by
     exact (Quadrilateral_cvx.nd₂₄ (QDR_cvx e.A e.B e.D e.C e.cvx_ABDC))
   --Because $AB = AC$ ,we have $\angle A B C = -\angle A C B$.
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_4.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_4.lean
index f717382d..37d261e3 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_4.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_4.lean
@@ -18,8 +18,8 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   C : Plane
   D : Plane
   --some nondegenerate
-  hnd₁ : ¬ collinear B C A
-  hnd₂ : ¬ collinear B C D
+  hnd₁ : ¬ Collinear B C A
+  hnd₂ : ¬ Collinear B C D
   --$A,D$ are on the opposite side of line $BC$,which satisfies $BD \para CA$(lemma needed), $BD = BC$
   BD_eq_BC : (SEG B D).length = (SEG B C).length
 instance B_ne_C {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.B e.C := ⟨(ne_of_not_collinear e.hnd₁).2.2.symm⟩
@@ -36,7 +36,7 @@ structure Setting2 (Plane : Type _) [EuclideanPlane Plane] extends Setting1 Plan
   E : Plane
   E_int : E LiesInt (SEG_nd B C)
   E_position : (SEG B E).length = (SEG A C).length
-lemma hnd₃ {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane}: ¬ collinear e.B e.E e.D := by sorry
+lemma hnd₃ {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane}: ¬ Collinear e.B e.E e.D := by sorry
 instance E_ne_D {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane} : PtNe e.E e.D := ⟨(ne_of_not_collinear hnd₃).1.symm⟩
 instance A_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane} : PtNe e.A e.B := ⟨(ne_of_not_collinear e.hnd₁).2.1.symm⟩
 instance B_ne_E {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane} : PtNe e.B e.E := ⟨(ne_of_not_collinear hnd₃).2.2.symm⟩
@@ -52,8 +52,8 @@ theorem Result {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane} : 
   We have $\triangle B E D \congr_a \triangle C A B$
   Thus $\angle B D E = -\angle C B A $.
   -/
-  have hnd₁' : ¬ collinear e.C e.A e.B := by
-    apply perm_collinear_trd_fst_snd.mt
+  have hnd₁' : ¬ Collinear e.C e.A e.B := by
+    apply Collinear.perm₃₁₂.mt
     exact e.hnd₁
   --$DB = BC$
   have e₂ : (SEG e.D e.B).length = (SEG e.B e.C).length := by
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_6.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_6.lean
index 7553546b..9dccc891 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_6.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_6.lean
@@ -13,11 +13,11 @@ In $▵ ABC$, $D, E$ are two different points on side $BC$, $AD = AE$, $∠ BAD
 Prove that $AB = AC$.
 -/
 -- Define $▵ ABC$.
-variable {A B C : P} {hnd1 : ¬ collinear A B C}
+variable {A B C : P} {hnd1 : ¬ Collinear A B C}
 -- $D, E$ are two different points on side $BC$.
 variable {D E : P} {hd : D LiesInt SEG B C} {he : E LiesInt SEG B C}
 -- nondegenerate
-lemma hnd2 : ¬ collinear A D E := by sorry
+lemma hnd2 : ¬ Collinear A D E := by sorry
 lemma a_ne_b : A ≠ B := by sorry
 lemma a_ne_d : A ≠ D := by sorry
 lemma a_ne_c : A ≠ C := by sorry
@@ -29,8 +29,8 @@ variable {hang : ∠ B A D a_ne_b a_ne_d = -∠ C A E a_ne_c a_ne_e}
 -- State the main goal.
 theorem Wuwowuji_Problem_1_6 : (SEG A B).length = (SEG A C).length := by
   -- nondegenerate
-  have hnd3 : ¬ collinear B D A := by sorry
-  have hnd4 : ¬ collinear C E A := by sorry
+  have hnd3 : ¬ Collinear B D A := by sorry
+  have hnd4 : ¬ Collinear C E A := by sorry
   -- Use ASA to prove $▵ BDA ≅ₐ ▵ CEA$.
   have h : (TRI_nd B D A hnd3) ≅ₐ (TRI_nd C E A hnd4) := by
     apply TriangleND.acongr_of_ASA
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_7.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_7.lean
index ba898093..79b27052 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_7.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_7.lean
@@ -15,7 +15,7 @@ Prove that $AD = BC$.
 -- $AD$ and $BC$ intersect at $O$.
 variable {A B C D O: P} {h1 : O LiesInt (SEG A D)} {h2 : O LiesInt (SEG B C)}
 -- nondegenerate
-variable {hnd1 : ¬ collinear C B A} {hnd2 : ¬ collinear D A B}
+variable {hnd1 : ¬ Collinear C B A} {hnd2 : ¬ Collinear D A B}
 lemma a_ne_b : A ≠ B := by sorry
 lemma a_ne_c : A ≠ C := by sorry
 lemma a_ne_d : A ≠ D := by sorry
@@ -33,7 +33,7 @@ theorem Wuwowuji_Problem_1_7 : (SEG A D).length = (SEG B C).length := by
   have o_ne_b : O ≠ B := by sorry
   have o_ne_c : O ≠ C := by sorry
   have o_ne_d : O ≠ D := by sorry
-  have hnd3 : ¬ collinear O A B := by sorry
+  have hnd3 : ¬ Collinear O A B := by sorry
   -- $▵ OAB$ is isoceles because $OA = OB$.
   have hisoc : (TRI_nd O A B hnd3).1.IsIsoceles := by
     calc
diff --git a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_8.lean b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_8.lean
index 67b80bd8..71f28a7d 100644
--- a/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_8.lean
+++ b/EuclideanGeometry/Example/Congruence_Exercise_ygr/Problem_8.lean
@@ -14,7 +14,7 @@ Prove that $▵ OPD ≅ₐ ▵ OPE$.
 -/
 
 -- Define $A, B, C, O$ on the plane.
-variable {A B C O : Plane} {hnd1 : ¬ collinear O A B} {hnd2 : ¬ collinear O B C} {hnd3 : ¬ collinear O C A}
+variable {A B C O : Plane} {hnd1 : ¬ Collinear O A B} {hnd2 : ¬ Collinear O B C} {hnd3 : ¬ Collinear O C A}
 -- nondegenerate
 lemma o_ne_a : O ≠ A:= by sorry
 lemma o_ne_b : O ≠ B := by sorry
@@ -26,8 +26,8 @@ variable {P : Plane} {hp : P LiesInt (SEG O C)}
 -- $D, E$ is the perpendicular foot from $P$ to line $OA, OB$.
 variable {D E : Plane} {hd : D = perp_foot P (LIN O A o_ne_a.symm)} {he : E = perp_foot P (LIN O B o_ne_b.symm)}
 -- State the main goal.
-lemma hnd4 : ¬ collinear O P D := by sorry
-lemma hnd5 : ¬ collinear O P E := by sorry
+lemma hnd4 : ¬ Collinear O P D := by sorry
+lemma hnd5 : ¬ Collinear O P E := by sorry
 theorem Wuwowuji_Problem_1_8 : (TRI_nd O P D hnd4) ≅ₐ (TRI_nd O P E hnd5) := by
   sorry
 
diff --git a/EuclideanGeometry/Example/ExerciseXT8.lean b/EuclideanGeometry/Example/ExerciseXT8.lean
index ad7ed15d..bf515faa 100644
--- a/EuclideanGeometry/Example/ExerciseXT8.lean
+++ b/EuclideanGeometry/Example/ExerciseXT8.lean
@@ -16,7 +16,7 @@ Prove that $\angle AY_1Z_2 + \angle BZ_1X_2 + \angle CX_1Y_2 + \angle Y_1Z_2A +
 -/
 
 -- Let $\triangle ABC$ be a nondegenerate triangle.
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 
 lemma c_ne_B : C ≠ B := sorry
 lemma B_ne_a : B ≠ A := sorry
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.1.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.1.lean
index 94e3e144..0c6cafd3 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.1.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.1.lean
@@ -16,7 +16,7 @@ Prove that $DM = EM$.
 -/
 /--/
 --Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {not_collinear_ABC: ¬ collinear A B C} {isoceles_ABC: (▵ A B C).IsIsoceles}
+variable {A B C : P} {not_collinear_ABC: ¬ Collinear A B C} {isoceles_ABC: (▵ A B C).IsIsoceles}
 --Let $D$ be a point on $AB$.
 variable {D : P} {D_on_seg: D LiesInt (SEG A B)}
 --Let $E$ be a point on $AC$
@@ -31,7 +31,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   isoc_ABC : (▵ A B C).IsIsoceles
   --Let $D$ be a point on $AB$.
   D : Plane
@@ -66,9 +66,9 @@ theorem Result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : (SE
       _ = (SEG e.C e.A).length := e.isoc_ABC.symm
       _ = (SEG e.A e.C).length := length_of_rev_eq_length'
   --Triangle B D M nondegenerate.
-  have h₁ : ¬ collinear e.B e.D e.M := by sorry
+  have h₁ : ¬ Collinear e.B e.D e.M := by sorry
   --Triangle C E M nondegenerate.
-  have h₂ : ¬ collinear e.C e.E e.M := by sorry
+  have h₂ : ¬ Collinear e.C e.E e.M := by sorry
   --Points not equal for the definition of angle is not invalid.
   --$D \ne B$ and $M \ne B$ for ∠ D B M.
   haveI d_ne_b : PtNe e.D e.B := ⟨ (ne_of_not_collinear h₁).2.2⟩
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.12.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.12.lean
index 352a6fca..b5bd0fee 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.12.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.12.lean
@@ -54,7 +54,7 @@ theorem SCHAUM_Problem_1_12 : Quadrilateral.IsParallelogram (QDR A B C D) := by
   sorry
   /-
   apply is_prg_of_para_eq_length'
-  · unfold perpendicular at *
+  · unfold Perpendicular at *
     unfold parallel
     have h : toProj (SegND B C (c_ne_B (hconv := hconv))) = toProj (SegND A D (A_ne_d (hconv := hconv)).symm) := by
       calc
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.14.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.14.lean
index dbd6ba64..baa4828c 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.14.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.14.lean
@@ -64,8 +64,8 @@ theorem result1 {Plane : Type _} [EuclideanPlane Plane] (e : Setting2 Plane) : (
   Thus $\triangle MBP \congr \triangle NCQ$ (by AAS),
   which implies $PM = QN$.
   -/
-  have not_collinear_MBP : ¬ collinear e.M e.B e.P := by sorry
-  have not_collinear_NDQ : ¬ collinear e.N e.D e.Q := by sorry
+  have not_collinear_MBP : ¬ Collinear e.M e.B e.P := by sorry
+  have not_collinear_NDQ : ¬ Collinear e.N e.D e.Q := by sorry
   have B_ne_M : e.B ≠ e.M := by sorry
   have P_ne_M : e.P ≠ e.M := by sorry
   have D_ne_N : e.D ≠ e.N := by sorry
@@ -85,7 +85,7 @@ theorem result2 {Plane : Type _} [EuclideanPlane Plane] (e : Setting2 Plane) : (
   -- We have $BD \perp QN$ because $N$ is the perpendicular foot from $Q$ to $BD$.
   have BD_perp_QN : LIN e.B e.D e.D_ne_B ⟂ LIN e.Q e.N e.N_ne_Q := by
     simp only [e.perp_foot_N]
-    exact perpendicular.symm (line_of_self_perp_foot_perp_line_of_not_lies_on e.not_Q_lieson_BD)
+    exact Perpendicular.symm (line_of_self_perp_foot_perp_line_of_not_lies_on e.not_Q_lieson_BD)
   -- then $PM \perp QN$ because $PM \perp BD$ and $BD \perp QN$.
   exact parallel_of_perp_perp (l₂ := (LIN e.B e.D e.D_ne_B)) PM_perp_BD BD_perp_QN
 
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.2'.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.2'.lean
index ed78d33a..d7dfb41a 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.2'.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.2'.lean
@@ -19,7 +19,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   isoceles_ABC : (▵ A B C).IsIsoceles
   --Let $D$ be a point on $AB$.
   D : Plane
@@ -46,7 +46,7 @@ structure Setting1 (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   hisoc : (▵ A B C).IsIsoceles
   --Let $D$ be a point on $AB$.
   D : Plane
@@ -118,7 +118,7 @@ theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : (SE
   -- We have $C \ne B$.
   have C_ne_B : e.C ≠ e.B := (ne_of_not_collinear e.not_collinear_ABC).1
   -- We have $\triangle PBD$ is nondegenerate
-  have not_collinear_PBD : ¬ collinear e.P e.B e.D := by sorry
+  have not_collinear_PBD : ¬ Collinear e.P e.B e.D := by sorry
   -- We have $B \ne D$.
   have B_ne_D : e.B ≠ e.D := (ne_of_not_collinear not_collinear_PBD).1.symm
   -- We have $P \ne D$.
@@ -126,7 +126,7 @@ theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : (SE
   -- We have $P \ne B$.
   have P_ne_B : e.P ≠ e.B := (ne_of_not_collinear not_collinear_PBD).2.2.symm
   -- We have $\triangle QCE$ is nondegenerate
-  have not_collinear_QCE : ¬ collinear e.Q e.C e.E := by sorry
+  have not_collinear_QCE : ¬ Collinear e.Q e.C e.E := by sorry
   -- We have $C \ne E$.
   have C_ne_E : e.C ≠ e.E := (ne_of_not_collinear not_collinear_QCE).1.symm
   -- We have $Q \ne E$.
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.2.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.2.lean
index e23d836e..e5d1e284 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.2.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.2.lean
@@ -15,7 +15,7 @@ Prove that $DP = EQ$.
 -/
 
 --Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : Plane} {not_collinear_ABC: ¬ collinear A B C} {isoceles_ABC: (▵ A B C).IsIsoceles}
+variable {A B C : Plane} {not_collinear_ABC: ¬ Collinear A B C} {isoceles_ABC: (▵ A B C).IsIsoceles}
 --Let $D$ be a point on $AB$.
 variable {D : Plane} {D_int_AB: D LiesInt (SEG A B)}
 --Let $E$ be a point on $AC$
@@ -96,7 +96,7 @@ theorem Problem1_2_ : (SEG D P).length = (SEG E Q).length := by
   -- We have $C \ne B$.
   have c_ne_b : C ≠ B := (ne_of_not_collinear not_collinear_ABC).1
   -- We have $\triangle PBD$ is nondegenerate
-  have not_collinear_ABC1 : ¬ collinear P B D := by sorry
+  have not_collinear_ABC1 : ¬ Collinear P B D := by sorry
   -- We have $B \ne D$.
   have b_ne_d : B ≠ D := (ne_of_not_collinear not_collinear_ABC1).1.symm
   -- We have $P \ne D$.
@@ -104,7 +104,7 @@ theorem Problem1_2_ : (SEG D P).length = (SEG E Q).length := by
   -- We have $P \ne B$.
   have p_ne_b : P ≠ B := (ne_of_not_collinear not_collinear_ABC1).2.2.symm
   -- We have $\triangle QCE$ is nondegenerate
-  have not_collinear_ABC2 : ¬ collinear Q C E := by sorry
+  have not_collinear_ABC2 : ¬ Collinear Q C E := by sorry
   -- We have $C \ne E$.
   have c_ne_e : C ≠ E := (ne_of_not_collinear not_collinear_ABC2).1.symm
   -- We have $Q \ne E$.
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.3.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.3.lean
index 7d44a017..9c6b6b83 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.3.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.3.lean
@@ -14,7 +14,7 @@ Prove that $∠DAB = ∠CAE$.
 -/
 /--/
 --Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {not_collinear_ABC: ¬ collinear A B C} {isoceles_ABC: (▵ A B C).IsIsoceles}
+variable {A B C : P} {not_collinear_ABC: ¬ Collinear A B C} {isoceles_ABC: (▵ A B C).IsIsoceles}
 --Let $D$ and $E$ be two points on $BC$
 variable {D : P} {D_Int_BC: D LiesInt (SEG B C)}
 variable {E : P} {E_Int_BC: E LiesInt (SEG B C)}
@@ -35,7 +35,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   isoc_ABC : (▵ A B C).IsIsoceles
   --Let $D$ and $E$ be two points on $BC$
   D : Plane
@@ -78,9 +78,9 @@ theorem Result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : ∠
       _= (SEG e.A e.B).length := length_of_rev_eq_length' --$BA = AB$ by symmetry
       _= (SEG e.C e.A).length := e.isoc_ABC.symm -- $AB = CA$ by isoceles.
   --Triangle B A D nondegenerate.
-  have hnd₁ : ¬ collinear e.B e.A e.D := by sorry
+  have hnd₁ : ¬ Collinear e.B e.A e.D := by sorry
   --Triangle C A E nondegenerate.
-  have hnd₂ : ¬ collinear e.C e.A e.E := by sorry
+  have hnd₂ : ¬ Collinear e.C e.A e.E := by sorry
   --Beacause $BD = CE$ we have $DB = EC$.
   have h₁ : (SEG e.D e.B).length = (SEG e.E e.C).length := by
     calc
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.4.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.4.lean
index b88db782..36557861 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.4.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.4.lean
@@ -21,12 +21,12 @@ variable {D : Plane} {D_on_seg: D LiesInt (SEG A F)}
 -- $AD = BF$.
 variable {seg_eq : (SEG A D).length = (SEG B F).length}
 --Let $C$ be a point.
-variable {C : Plane} {C_off_lin: ¬ collinear A F C} --Implied by opposite side.
+variable {C : Plane} {C_off_lin: ¬ Collinear A F C} --Implied by opposite side.
 --Let $E$ be a point on the opposite side of $AF$ to $C$, such that EF ∥ AC and ED ∥ BC.
-variable {E : Plane} {E_off_lin: ¬ collinear A F E} --Implied by opposite side.
+variable {E : Plane} {E_off_lin: ¬ Collinear A F E} --Implied by opposite side.
 -- Claim:$C \ne A$ , $C \ne B$, $C, A, B$ is not collinear, $E, F, D$ is not collinear.
-lemma cabnd : ¬ collinear C A B := by sorry
-lemma efdnd : ¬ collinear E F D := by sorry
+lemma cabnd : ¬ Collinear C A B := by sorry
+lemma efdnd : ¬ Collinear E F D := by sorry
 lemma c_ne_a : C ≠ A := (ne_of_not_collinear C_off_lin).2.1.symm
 lemma c_ne_B : C ≠ B :=by
   by_contra h
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.5.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.5.lean
index 70e034d8..1d87ffc6 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.5.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.5.lean
@@ -79,9 +79,9 @@ As a consequence, we know that $PQRS$ is a parallelogram.
   -- We have $C \ne B$.
   have C_ne_B : e.C ≠ e.B := by sorry
   -- We have that $A, P, C$ is not collinear.
-  have not_collinear_APC : ¬ collinear e.A e.P e.C := by sorry
+  have not_collinear_APC : ¬ Collinear e.A e.P e.C := by sorry
   -- We have that $A, S, C$ is not collinear.
-  have not_collinear_ASC : ¬ collinear e.A e.S e.C := by sorry
+  have not_collinear_ASC : ¬ Collinear e.A e.S e.C := by sorry
   -- We have that $S$ lies on $AD$ by applying symmetry to the fact that $S$ lies on $DA$.
   have S_int_AD : e.S LiesInt (SEG e.A e.D) := Seg.lies_int_rev_iff_lies_int.mp e.S_int_DA
   -- We have that $Q$ lies on $CB$ by applying symmetry to the fact that $Q$ lies on $BC$.
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.7.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.7.lean
index e055f103..4e980dda 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.7.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.7.lean
@@ -23,7 +23,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   isoceles_ABC : (▵ A B C).IsIsoceles
 --Let $D$ be point on the segment $BC$
   D : Plane
@@ -105,7 +105,7 @@ Therefore, $DX = EY$.
   -- We have $Y \ne C$ as Y is on the interior of ray $CA$ and therefore different to the source $C$.
   have Y_ne_C : e.Y ≠ e.C := Y_int_ray_CA.2
   -- We have $Y, C, E$ are not collinear because $E$ doesn't lies on line $AC$ and $C$ doesn't coincide with $Y$.
-  have not_collinear_YCE : ¬ collinear e.Y e.C e.E := by exact not_collinear_with_perp_foot_of_ne_perp_foot e.E e.C e.Y (LIN e.A e.C e.C_ne_A) (Line.snd_pt_lies_on_mk_pt_pt e.C_ne_A) E_not_on_AC e.he (Y_ne_C).symm
+  have not_collinear_YCE : ¬ Collinear e.Y e.C e.E := by exact not_collinear_with_perp_foot_of_ne_perp_foot e.E e.C e.Y (LIN e.A e.C e.C_ne_A) (Line.snd_pt_lies_on_mk_pt_pt e.C_ne_A) E_not_on_AC e.he (Y_ne_C).symm
   -- We have $E \ne Y$.
   have E_ne_Y : e.E ≠ e.Y := by sorry
 
@@ -135,7 +135,7 @@ Therefore, $DX = EY$.
   -- We have $X \ne B$ as X is on the interior of ray $BA$ and therefore different to the source $B$.
   have X_ne_B : e.X ≠ e.B := X_int_ray_BA.2
   -- We have $X, B, D$ are not collinear because $D$ doesn't lies on line $AB$ and $B$ doesn't coincide with $X$.
-  have not_collinear_XBD : ¬ collinear e.X e.B e.D := by
+  have not_collinear_XBD : ¬ Collinear e.X e.B e.D := by
     exact not_collinear_with_perp_foot_of_ne_perp_foot e.D e.B e.X (LIN e.A e.B e.B_ne_A) (Line.snd_pt_lies_on_mk_pt_pt e.B_ne_A) D_not_on_AB e.hd (X_ne_B).symm
   -- We have $D \ne X$.
   have D_ne_X : e.D ≠ e.X := by sorry
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.8.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.8.lean
index 3eabf5d9..9d238f9f 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.8.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.8.lean
@@ -21,7 +21,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   regular_ABC : (▵ A B C).IsRegular
 --Claim $A \ne B$
   A_ne_B : A ≠ B :=
@@ -61,9 +61,9 @@ Thus, $\triangle BDA \cong \triangle CEB$ (by SAS).
 Therefore, $BD = CE$.
 -/
   --We have that $B, D, A$ is not collinear.
-  have not_collinear_BDA : ¬ collinear e.B e.D e.A := by sorry
+  have not_collinear_BDA : ¬ Collinear e.B e.D e.A := by sorry
   --We have that $C, E, B$ is not collinear.
-  have not_collinear_CEB : ¬ collinear e.C e.E e.B := by sorry
+  have not_collinear_CEB : ¬ Collinear e.C e.E e.B := by sorry
   --We have $D \ne B$.
   have D_ne_B : e.D ≠ e.B := by sorry
   --We have $A \ne B$.
diff --git a/EuclideanGeometry/Example/SCHAUM/Problem1.9.lean b/EuclideanGeometry/Example/SCHAUM/Problem1.9.lean
index 7bce34f2..94021f15 100644
--- a/EuclideanGeometry/Example/SCHAUM/Problem1.9.lean
+++ b/EuclideanGeometry/Example/SCHAUM/Problem1.9.lean
@@ -22,7 +22,7 @@ structure Setting1 (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   isoceles_ABC : (▵ A B C).IsIsoceles
 -- Claim $B \ne A$.
   B_ne_A : B ≠ A :=
diff --git a/EuclideanGeometry/Example/ShanZun-Ex1a.lean b/EuclideanGeometry/Example/ShanZun-Ex1a.lean
index c7ff5978..892148a3 100644
--- a/EuclideanGeometry/Example/ShanZun-Ex1a.lean
+++ b/EuclideanGeometry/Example/ShanZun-Ex1a.lean
@@ -13,7 +13,7 @@ namespace Shan_Problem_1_3
 
 Prove that $CD = 2 \cdot CE$. -/
   -- Define an isosceles triangle ABC
-  variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+  variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
   -- A ≠ B
   lemma A_ne_B : A ≠ B := (ne_of_not_collinear hnd).2.2.symm
   -- Since D is on line AB and AB = BD, it is trivial that D is on ray AB. Position can be determined here.
@@ -23,8 +23,8 @@ Prove that $CD = 2 \cdot CE$. -/
   variable {E : P} {E_midpoint : E = (SEG A B).midpoint}
   -- State the main goal
   theorem Shan_Problem_1_3 : (SEG C D).length = 2 * (SEG C E).length := by
-    have h_AEC_nd : ¬ collinear A E C := by sorry
-    have h_ACD_nd : ¬ collinear A C D := by sorry
+    have h_AEC_nd : ¬ Collinear A E C := by sorry
+    have h_ACD_nd : ¬ Collinear A C D := by sorry
     have h_AC_over_AE : (SEG A C).length / (SEG A E).length = 2 := by sorry
     have h_AD_over_AC : (SEG A D).length / (SEG A C).length = 2 := by sorry
     have h_sim: (TRI_nd A E C h_AEC_nd) ∼ₐ (TRI_nd A C D h_ACD_nd) := sorry
@@ -39,7 +39,7 @@ namespace Shan_Problem_1_4
 Prove that $BD = AC + CD$.-/
 
   -- Define a triangle ABC
-  variable {A B C : P} {hnd : ¬ collinear A B C}
+  variable {A B C : P} {hnd : ¬ Collinear A B C}
   -- Possibly needed non-linear conditions
   lemma A_ne_B : A ≠ B := (ne_of_not_collinear hnd).2.2.symm
   lemma A_ne_C : A ≠ C := (ne_of_not_collinear hnd).2.1
diff --git a/EuclideanGeometry/Example/ShanZun-Ex1b.lean b/EuclideanGeometry/Example/ShanZun-Ex1b.lean
index 4ca3a262..2ced19d9 100644
--- a/EuclideanGeometry/Example/ShanZun-Ex1b.lean
+++ b/EuclideanGeometry/Example/ShanZun-Ex1b.lean
@@ -14,7 +14,7 @@ namespace Shan_Problem_1_5
 Prove that $AF = EF$. -/
 
   -- Let $\triangle ABC$ be an triangle.
-  variable {A B C : P} {hnd : ¬ collinear A B C}
+  variable {A B C : P} {hnd : ¬ Collinear A B C}
   lemma B_ne_C : B ≠ C := (ne_of_not_collinear hnd).1.symm
   variable {D : P} {median_D_position : D = (SEG B C).midpoint}
   variable {median : SegND P} {defmedian: median = (SEG A D)}
@@ -30,7 +30,7 @@ namespace Shan_Problem_1_6
 
 Prove that For any point $D$ on the base $BC$, the sum of the the distance of $D$ to $AB$ and to $AC$ is independent of $D$. -/
   -- Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-  variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+  variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
   -- Claim: $A \ne B$ and $A \neq C$. This is because vertices of nondegenerate triangles are distinct.
   lemma B_ne_a : B ≠ A := (ne_of_not_collinear hnd).2.2
   lemma c_ne_a : C ≠ A := (ne_of_not_collinear hnd).2.1.symm
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1.lean b/EuclideanGeometry/Example/ShanZun/Ex1.lean
index 8db822f4..1accc03e 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1.lean
@@ -13,7 +13,7 @@ namespace Shan_Problem_1_1
 Prove that $\angle DAE = (\angle CBA - \angle ACB) / 2$. -/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -50,7 +50,7 @@ such that $\angle EBC = \angle BCD$.
 Prove that $\angle CDA = \angle BEA$. -/
 
 -- Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
 -- Claim: $B \ne A$ and $C \ne A$ and $B \ne C$. This is because vertices of nondegenerate triangles are distinct.
 lemma B_ne_a : B ≠ A := (ne_of_not_collinear hnd).2.2
 lemma c_ne_a : C ≠ A := (ne_of_not_collinear hnd).2.1.symm
@@ -59,7 +59,7 @@ lemma B_ne_C : B ≠ C := (ne_of_not_collinear hnd).1.symm
 variable {D E : P} {hd : D LiesInt (SEG_nd A B (B_ne_a (hnd := hnd))).extension} {he : E LiesInt (SEG_nd A C (c_ne_a (hnd := hnd))).extension}
 -- Claim: $E \ne B$ and $D \ne C$. This is because $E$ lies on line $AC$, but $B$ doesn't lies on $AC$; $D$ lies on line $AB$, but $C$ doesn't lies on $AB$.
 lemma e_ne_B : E ≠ B := by -- for $E \ne B$
-  have b_not_lieson_ac := (Line.lies_on_line_of_pt_pt_iff_collinear (_h := ⟨ c_ne_a (hnd := hnd)⟩) B).mp.mt (flip_collinear_snd_trd.mt hnd) -- $B$ doesn't lies on line $AC$ because $A, B, C$ not collinear
+  have b_not_lieson_ac := (Line.lies_on_line_of_pt_pt_iff_collinear (_h := ⟨ c_ne_a (hnd := hnd)⟩) B).mp.mt (Collinear.perm₁₃₂.mt hnd) -- $B$ doesn't lies on line $AC$ because $A, B, C$ not collinear
   have e_lieson_ac := SegND.lies_on_toLine_of_lies_on_extn (Ray.lies_on_of_lies_int he) -- $E$ lieson line $AC$ because $E$ lies in the extension of $AC$
   exact ne_of_lieson_and_not_lieson e_lieson_ac b_not_lieson_ac -- $E \ne B$ because $E$ lies on line $AC$, but $B$ doesn't lies on line $AC$
 lemma d_ne_c : D ≠ C := by -- for $D \ne C$
@@ -89,9 +89,9 @@ theorem Shan_Problem_1_2 : ∠ C D A (d_ne_c (hnd := hnd) (hd := hd)).symm (d_ne
   -- $\angle CBD = \angle ECB$
   have ang₂ : ∠ C B D (B_ne_C (hnd := hnd)).symm d_ne_B = ∠ E C B e_ne_C (B_ne_C (hnd := hnd)) := sorry
   -- $C,B,D$ are not collinear
-  have not_coli_cbd : ¬ collinear C B D := sorry
+  have not_coli_cbd : ¬ Collinear C B D := sorry
   -- $B,C,E$ are not collinear
-  have not_coli_bec : ¬ collinear B C E := sorry
+  have not_coli_bec : ¬ Collinear B C E := sorry
   -- $\triangle CBD \cong \triangle BCE$
   have iso : (TRI_nd C B D not_coli_cbd) IsCongrTo (TRI_nd B C E not_coli_bec) := sorry
   -- $\angle CDB == \angle CDA$
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1a.lean b/EuclideanGeometry/Example/ShanZun/Ex1a.lean
index d89b6965..18510ab5 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1a.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1a.lean
@@ -14,7 +14,7 @@ namespace Shan_Problem_1_3
 Prove that $CD = 2 \cdot CE$. -/
 
 -- Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -47,7 +47,7 @@ namespace Shan_Problem_1_4
 Prove that $BD = AC + CD$.-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -66,7 +66,7 @@ theorem Shan_Problem_1_4 : (SEG B D).length = (SEG A C).length + (SEG C D).lengt
   -- $DE = AC + CD$
   have de_eq_ac_plus_cd : (SEG D E).length = (SEG A C).length + (SEG C D).length := sorry
   -- $C A E$ is not collinear
-  have cae_not_collinear : ¬ collinear C A E := sorry
+  have cae_not_collinear : ¬ Collinear C A E := sorry
   -- $\triangle CAE$ is isoceles
   have isoceles₁ : (TRI_nd C A E cae_not_collinear).1.IsIsoceles := by
     rw[← Seg.length_of_rev_eq_length (seg := (SEG C E))] at ce_eq_ca
@@ -83,7 +83,7 @@ theorem Shan_Problem_1_4 : (SEG B D).length = (SEG A C).length + (SEG C D).lengt
   -- $\angle EBA = \angle AEB$
   have ang₃ : ∠ E B A e_ne_b A_ne_B = ∠ A E B e_ne_a.symm e_ne_b.symm := sorry
   -- $ABE$ is not collinear
-  have abe_not_collinear : ¬ collinear A B E := sorry
+  have abe_not_collinear : ¬ Collinear A B E := sorry
   -- $\triangle ABE$ is isoceles
   have isoceles₂ : (TRI_nd A B E abe_not_collinear).1.IsIsoceles := by
     apply is_isoceles_tri_iff_ang_eq_ang_of_nd_tri.mpr
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1b.lean b/EuclideanGeometry/Example/ShanZun/Ex1b.lean
index f7b1cfd1..910259cf 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1b.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1b.lean
@@ -14,7 +14,7 @@ namespace Shan_Problem_1_5
 Prove that $AF = EF$. -/
 
 -- Let $\triangle ABC$ be an triangle.
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 def tri : Triangle P := ▵ A B C
 -- Claim: $B \ne C$. This is because vertices of nondegenerate triangles are distinct.
 lemma A_ne_C : A ≠ C := (ne_of_not_collinear hnd).2.1
@@ -52,7 +52,7 @@ Prove that for any point $D$ on the base $BC$,
 the sum of the the distance of $D$ to $AB$ and to $AC$ is independent of $D$. -/
 
 -- Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1c'.lean b/EuclideanGeometry/Example/ShanZun/Ex1c'.lean
index 57274653..a972254f 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1c'.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1c'.lean
@@ -18,7 +18,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Claim :$C \ne A$
   C_ne_A : C ≠ A :=
     -- This is because vertices $A, C$ of a nondegenerate triangle are distinct.
@@ -52,12 +52,12 @@ theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : (SE
     rw[hE]
     apply (SegND.midpt_lies_int (seg_nd := SEG_nd e.A e.C e.C_ne_A)).2.2
   --midpoint lies on the segment
-  have adb_collinear : collinear e.A e.D e.B := by
+  have adb_collinear : Collinear e.A e.D e.B := by
     apply collinear_of_vec_eq_smul_vec'
     use 2
     simp only [e.hD,Seg.midpoint,one_div, seg_toVec_eq_vec, vec_of_pt_vadd_pt_eq_vec,smul_smul]
     norm_num
-  have aec_collinear : collinear e.A E e.C := by
+  have aec_collinear : Collinear e.A E e.C := by
     apply collinear_of_vec_eq_smul_vec'
     use 2
     simp only [hE,Seg.midpoint,one_div, seg_toVec_eq_vec, vec_of_pt_vadd_pt_eq_vec,smul_smul]
@@ -88,26 +88,26 @@ theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) : (SE
     exact hE
     rw[hE]
     apply Seg.midpt_lies_on
-  have not_collinear_ADE: ¬ collinear e.A e.D E := by
+  have not_collinear_ADE: ¬ Collinear e.A e.D E := by
     intro h'
-    have : collinear e.A e.B E := by
+    have : Collinear e.A e.B E := by
       apply collinear_of_collinear_collinear_ne adb_collinear h' D_ne_A
-    have neghnd : collinear e.A e.B e.C := by
-      apply collinear_of_collinear_collinear_ne (flip_collinear_snd_trd this) aec_collinear E_ne_A
+    have neghnd : Collinear e.A e.B e.C := by
+      apply collinear_of_collinear_collinear_ne (Collinear.perm₁₃₂ this) aec_collinear E_ne_A
     exact e.not_collinear_ABC neghnd
-  have not_collinear_CDE : ¬ collinear e.C e.D E := by
+  have not_collinear_CDE : ¬ Collinear e.C e.D E := by
     intro h
-    have : collinear e.C e.D e.A := by
-      apply flip_collinear_snd_trd
+    have : Collinear e.C e.D e.A := by
+      apply Collinear.perm₁₃₂
       apply collinear_of_collinear_collinear_ne
-      apply (flip_collinear_snd_trd (flip_collinear_fst_snd (flip_collinear_snd_trd aec_collinear)))
-      apply flip_collinear_snd_trd h
+      apply (Collinear.perm₁₃₂ (Collinear.perm₂₁₃ (Collinear.perm₁₃₂ aec_collinear)))
+      apply Collinear.perm₁₃₂ h
       apply E_ne_C
-    have : collinear e.A e.B e.C := by
+    have : Collinear e.A e.B e.C := by
       apply collinear_of_collinear_collinear_ne
       apply adb_collinear
       -- apply e.hD
-      apply (flip_collinear_snd_trd (flip_collinear_fst_snd (flip_collinear_snd_trd this)))
+      apply (Collinear.perm₁₃₂ (Collinear.perm₂₁₃ (Collinear.perm₁₃₂ this)))
       apply D_ne_A
     apply e.not_collinear_ABC this
 
@@ -184,7 +184,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Claim :$C \ne A$
   C_ne_A : C ≠ A :=
     -- This is because vertices $A, C$ of a nondegenerate triangle are distinct.
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1c.lean b/EuclideanGeometry/Example/ShanZun/Ex1c.lean
index e0e51207..5676b29a 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1c.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1c.lean
@@ -13,7 +13,7 @@ namespace Shan_Problem_1_7
 
 Prove that $CD = AB / 2$. -/
 
-variable {A B C: P} {hnd : ¬ collinear A B C}
+variable {A B C: P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $A \ne C$ and $B \ne C$.
 -- This is because vertices of nondegenerate triangles are distinct.
 lemma B_ne_a : B ≠ A := (ne_of_not_collinear hnd).2.2
@@ -28,8 +28,8 @@ lemma d_ne_a: D ≠ A := by
   apply (SegND.midpt_lies_int (SegND := SegND A B (b_ne_a).symm)).2.1
   use C
   by_contra h
-  have : collinear A B C :=by
-    apply flip_collinear_fst_snd h
+  have : Collinear A B C :=by
+    apply Collinear.perm₂₁₃ h
   trivial
 --Introduce the midpoint E of AC
 variable {E : P} {he : E=  (SEG A C).midpoint}
@@ -44,12 +44,12 @@ lemma e_ne_C: E ≠ C := by
   use B
   exact hnd
 --midpoint lies on the segment
-lemma adb_collinear : collinear A D B := by
+lemma adb_collinear : Collinear A D B := by
   apply collinear_of_vec_eq_smul_vec'
   use 2
   simp only [hd,Seg.midpoint,one_div, seg_toVec_eq_vec, vec_of_pt_vadd_pt_eq_vec,smul_smul]
   norm_num
-lemma aec_collinear : collinear A E C := by
+lemma aec_collinear : Collinear A E C := by
   apply collinear_of_vec_eq_smul_vec'
   use 2
   simp only [he,Seg.midpoint,one_div, seg_toVec_eq_vec, vec_of_pt_vadd_pt_eq_vec,smul_smul]
@@ -82,40 +82,40 @@ lemma ae_ratio : (SEG A E).length / (SEG A C).length = 2⁻¹ :=by
   rw[he]
   apply Seg.midpt_lies_on
 
-lemma hnd': ¬ collinear A D E := by
+lemma hnd': ¬ Collinear A D E := by
   intro h'
-  have : collinear A B E := by
+  have : Collinear A B E := by
     apply collinear_of_collinear_collinear_ne adb_collinear h' d_ne_a
     exact hd
     use B
     use C
     exact hnd
     exact hd
-  have neghnd : collinear A B C := by
-    apply collinear_of_collinear_collinear_ne (flip_collinear_snd_trd this) aec_collinear e_ne_a
+  have neghnd : Collinear A B C := by
+    apply collinear_of_collinear_collinear_ne (Collinear.perm₁₃₂ this) aec_collinear e_ne_a
     exact he
     use B
     use C
     exact hnd
     exact he
   apply hnd neghnd
-lemma hnd'' : ¬ collinear C D E := by
+lemma hnd'' : ¬ Collinear C D E := by
   intro h
-  have : collinear C D A := by
-    apply flip_collinear_snd_trd
+  have : Collinear C D A := by
+    apply Collinear.perm₁₃₂
     apply collinear_of_collinear_collinear_ne
-    apply (flip_collinear_snd_trd (flip_collinear_fst_snd (flip_collinear_snd_trd aec_collinear)))
+    apply (Collinear.perm₁₃₂ (Collinear.perm₂₁₃ (Collinear.perm₁₃₂ aec_collinear)))
     use E
     exact he
-    apply flip_collinear_snd_trd h
+    apply Collinear.perm₁₃₂ h
     apply e_ne_C
     apply hnd
     exact he
-  have : collinear A B C := by
+  have : Collinear A B C := by
     apply collinear_of_collinear_collinear_ne
     apply adb_collinear
     apply hd
-    apply (flip_collinear_snd_trd (flip_collinear_fst_snd (flip_collinear_snd_trd this)))
+    apply (Collinear.perm₁₃₂ (Collinear.perm₂₁₃ (Collinear.perm₁₃₂ this)))
     apply d_ne_a
     apply hnd
     exact hd
@@ -210,7 +210,7 @@ namespace Shan_Problem_1_8
 /- In $\triangle ABC$, let $BD$ and $CE$ be the heights, with foots $D$ and $E$, respectively. Let $F$ and $G$ be the midpoint of $BC$ and $DE$, respectively.
 
 Prove that $FG \perp DE$. -/
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1d.lean b/EuclideanGeometry/Example/ShanZun/Ex1d.lean
index 4e37365a..3785e5ee 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1d.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1d.lean
@@ -14,7 +14,7 @@ namespace Shan_Problem_1_9
 Prove that $AB = 2\cdot DE$. -/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma a_ne_b : A ≠ B := sorry
 lemma b_ne_c : B ≠ C := sorry
@@ -49,7 +49,7 @@ theorem Shan_Problem_1_9 : (SEG A B).length = 2 * (SEG D E).length := by
   -- $\angle DFE = \angle FED$
   have ang₄ : ∠ E F D f_ne_e.symm f_ne_d.symm = ∠ D E F d_ne_e f_ne_e := sorry
   -- $D F E$ is not collinear
-  have dfe_not_collinear : ¬ collinear D F E := sorry
+  have dfe_not_collinear : ¬ Collinear D F E := sorry
   -- $ED = DF$ because $\triangle DFE$ is isoceles
   have dfe_isoceles : (TRI_nd D F E dfe_not_collinear).1.IsIsoceles := by
     apply is_isoceles_tri_iff_ang_eq_ang_of_nd_tri.mpr
@@ -68,7 +68,7 @@ namespace Shan_Problem_1_10
 Prove that $AD = DC$. -/
 
 -- Let $\triangle ABC$ be an isosceles triangle in which $AB = AC$.
-variable {A B C : P} {hnd : ¬ collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {isoceles_ABC : (▵ A B C).IsIsoceles}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma a_ne_b : A ≠ B := sorry
 lemma b_ne_c : B ≠ C := sorry
@@ -116,7 +116,7 @@ theorem Shan_Problem_1_10 : (SEG A D).length = (SEG D C).length := by
   -- $\angle BMC = 2\pi / 5$
   have ang₁ : ∠ B M C b_ne_m m_ne_c.symm = ↑ (2 * π / 5) := sorry
   -- $BAM$ is not collinear
-  have bam_not_collinear : ¬ collinear B A M := sorry
+  have bam_not_collinear : ¬ Collinear B A M := sorry
   -- $\triangle BAM$ is isoceles
   have isocele₁ : (TRI_nd B A M bam_not_collinear).1.IsIsoceles := sorry
   -- Let $N$ be the midpoint of $AC$
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1e'.lean b/EuclideanGeometry/Example/ShanZun/Ex1e'.lean
index 6bdf39da..3524919c 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1e'.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1e'.lean
@@ -15,7 +15,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Claim : $B \ne A$
   B_ne_A : B ≠ A :=
     -- This is because vertices $B, C$ of a nondegenerate triangle are distinct.
@@ -57,7 +57,7 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Claim :$C \ne A$
   C_ne_A : C ≠ A :=
     -- This is because vertices $A, C$ of a nondegenerate triangle are distinct.
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1e.lean b/EuclideanGeometry/Example/ShanZun/Ex1e.lean
index 065f408e..d3af9003 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1e.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1e.lean
@@ -12,7 +12,7 @@ the perpendicular line passing through $D$ of the bisector of $\angle BAC$ inter
 Prove that $BE=CF=\frac{1}{2}|AB-AC|$.  -/
 
 -- We have triagngle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -34,7 +34,7 @@ namespace Shan_Problem_2_18
 Prove that $E$ liesInt $DF$ -/
 
 -- We have triagngle $\triangle ABC$ such that $AC < BC$
-variable {A B C : P} {hnd : ¬ collinear A B C} {hedge : (SEG A C).length < (SEG B C).length}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {hedge : (SEG A C).length < (SEG B C).length}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -56,7 +56,7 @@ namespace Shan_Problem_2_18'
 If $\angle ACB = 90^{circ}$, prove that $CE$ is the bisector of $\angle FCD$.  -/
 
 -- We have triagngle $\triangle ABC$ such that $AC < BC$
-variable {A B C : P} {hnd : ¬ collinear A B C} {hedge : (SEG A C).length < (SEG B C).length}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {hedge : (SEG A C).length < (SEG B C).length}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1f'.lean b/EuclideanGeometry/Example/ShanZun/Ex1f'.lean
index fbb48aff..db9d0575 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1f'.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1f'.lean
@@ -18,7 +18,7 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  hnd : ¬ collinear A B C
+  hnd : ¬ Collinear A B C
   hreg : (▵ A B C).IsRegular
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 instance A_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.A e.B := ⟨(ne_of_not_collinear e.hnd).2.2.symm⟩
@@ -115,8 +115,8 @@ theorem Result {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane} : (S
     right
     exact sixty₂
   -- $BFE$ is not collinear because $\angle FBE = \frac{\pi}{3}$ or $ - \frac{\pi}{3}$
-  have BFE_not_collinear : ¬ collinear e.B e.F e.E := sorry
-  have BEF_not_collinear : ¬ collinear e.B e.E e.F := sorry
+  have BFE_not_collinear : ¬ Collinear e.B e.F e.E := sorry
+  have BEF_not_collinear : ¬ Collinear e.B e.E e.F := sorry
   -- $\triangle BFE$ is regular because $BF = EB$ and $\angle FBE = \frac{\pi}{3}$ or $ - \frac{\pi}{3}$
   have BFE_is_regular : (TRI_nd e.B e.F e.E BFE_not_collinear).1.IsRegular := by
     apply regular_tri_of_isoceles_tri_of_fst_ang_eq_sixty_deg
@@ -161,8 +161,8 @@ theorem Result {Plane : Type _} [EuclideanPlane Plane] {e : Setting2 Plane} : (S
     _ = (SEG e.B e.E).length := by
       simp only [length_of_rev_eq_length']
   -- $BEC$ is not collinear and $FED$ is not collinear
-  have BEC_not_collinear : ¬ collinear e.B e.E e.C := sorry
-  have FED_not_collinear : ¬ collinear e.F e.E e.D := sorry
+  have BEC_not_collinear : ¬ Collinear e.B e.E e.C := sorry
+  have FED_not_collinear : ¬ Collinear e.F e.E e.D := sorry
   -- $\triangle BCE$ is anti-congruence to $\triangle FDE$
   have cong : (TRI_nd e.B e.E e.C BEC_not_collinear) ≅ₐ (TRI_nd e.F e.E e.D FED_not_collinear) := TriangleND.acongr_of_SAS DF_eq_CB.symm ang_EBC_eq_neg_ang_EFD FE_eq_BE.symm
   -- $EC = ED$ because $\triangle BCE \cong \triangle FDE$
@@ -182,7 +182,7 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  hnd : ¬ collinear A B C
+  hnd : ¬ Collinear A B C
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 instance A_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.A e.B := ⟨(ne_of_not_collinear e.hnd).2.2.symm⟩
 instance B_ne_C {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.B e.C := ⟨(ne_of_not_collinear e.hnd).1.symm⟩
@@ -226,7 +226,7 @@ structure Setting1  (Plane : Type _) [EuclideanPlane Plane] where
   A : Plane
   B : Plane
   C : Plane
-  hnd : ¬ collinear A B C
+  hnd : ¬ Collinear A B C
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 instance A_ne_B {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.A e.B := ⟨(ne_of_not_collinear e.hnd).2.2.symm⟩
 instance B_ne_C {Plane : Type _} [EuclideanPlane Plane] {e : Setting1 Plane} : PtNe e.B e.C := ⟨(ne_of_not_collinear e.hnd).1.symm⟩
@@ -269,7 +269,7 @@ such that $AE = 2 EC$
 Prove that $AF = 3 FB$ -/
 
 -- We have acute triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C} {hacute : TriangleND.IsAcute (TRI_nd A B C hnd)}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {hacute : TriangleND.IsAcute (TRI_nd A B C hnd)}
 -- 这个题应该需要加锐角三角形的限制，否则需要条件中的$AE = 2 EC$是有向线段的相等
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
@@ -292,7 +292,7 @@ let the angle bisectors of $\angle ADB$ and $\angle ADC$ intersect $AB$ and $AC$
 Prove that $EF \parallel BC$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -320,7 +320,7 @@ $CD,BE$ intersects at $O$
 Prove that $OE = \frac{1}{4} BE$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- $D$ is midpoint of $AB$
 variable {D : P} {hd : D = (SEG A B).midpoint}
 -- $E$ lies on $AC$ such that $AE = 2 CE$,
@@ -339,7 +339,7 @@ The parallel line to $AC$ of $E,F$ intersect $BC$ at $G,H$ respectively,
 Prove that $EG + FH = AC$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -396,7 +396,7 @@ the angle bisector of $\angle ABC$ intersect $AD$ and $AC$ at $M,N$ respectively
 Prove that $AB^2 - AN^2 = BM \times BN$-/
 
 -- Let triangle $\triangle ABC$ be a right triangle with $\angle BAC = 90^{circ}$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma a_ne_b : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1f.lean b/EuclideanGeometry/Example/ShanZun/Ex1f.lean
index b6e2b0aa..905be2b9 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1f.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1f.lean
@@ -14,7 +14,7 @@ such that $AE = BD$, connect $CE,DE$
 Prove that $CE = DE$ -/
 
 -- Let $\triangle ABC$ be a regular triangle
-variable {A B C : P} {hnd : ¬ collinear A B C} {hreg : (▵ A B C).IsRegular}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {hreg : (▵ A B C).IsRegular}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -121,7 +121,7 @@ theorem Shan_Problem_2_11 : (SEG C E).length = (SEG D E).length := by
     right
     exact sixty₂
   -- $BFE$ is not collinear
-  have BFE_not_collinear : ¬ collinear B F E := sorry
+  have BFE_not_collinear : ¬ Collinear B F E := sorry
   -- $\triangle BFE$ is regular
   have BFE_is_regular : (TRI_nd B F E BFE_not_collinear).1.IsRegular := by
     apply regular_tri_of_isoceles_tri_of_fst_ang_eq_sixty_deg
@@ -146,8 +146,8 @@ theorem Shan_Problem_2_11 : (SEG C E).length = (SEG D E).length := by
     _ = (SEG B E).length := by
       simp only [length_of_rev_eq_length']
   -- $BEC$ is not collinear and $FED$ is not collinear
-  have BEC_not_collinear : ¬ collinear B E C := sorry
-  have FED_not_collinear : ¬ collinear F E D := sorry
+  have BEC_not_collinear : ¬ Collinear B E C := sorry
+  have FED_not_collinear : ¬ Collinear F E D := sorry
   -- $\triangle BCE$ is anti-congruence to $\triangle FDE$
   have cong : (TRI_nd B E C BEC_not_collinear) ≅ₐ (TRI_nd F E D FED_not_collinear) := TriangleND.acongr_of_SAS DF_eq_CB.symm ang_EBC_eq_neg_ang_EFD FE_eq_BE.symm
   -- $EC = ED$ because $\triangle BCE \cong \triangle FDE$
@@ -163,7 +163,7 @@ line $DF$ and $EG$ intersect at $H$
 Prove that quadrilateral $ABCH$ is parallelogram -/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -191,7 +191,7 @@ such that $FB = CG$ and $AF \parallel BE$ ,
 Prove that $AG \parallel DC$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -224,7 +224,7 @@ such that $AE = 2 EC$
 Prove that $AF = 3 FB$ -/
 
 -- We have acute triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C} {hacute : TriangleND.IsAcute (TRI_nd A B C hnd)}
+variable {A B C : P} {hnd : ¬ Collinear A B C} {hacute : TriangleND.IsAcute (TRI_nd A B C hnd)}
 -- 这个题应该需要加锐角三角形的限制，否则需要条件中的$AE = 2 EC$是有向线段的相等
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
@@ -247,7 +247,7 @@ let the angle bisectors of $\angle ADB$ and $\angle ADC$ intersect $AB$ and $AC$
 Prove that $EF \parallel BC$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -275,7 +275,7 @@ $CD,BE$ intersects at $O$
 Prove that $OE = \frac{1}{4} BE$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- $D$ is midpoint of $AB$
 variable {D : P} {hd : D = (SEG A B).midpoint}
 -- $E$ lies on $AC$ such that $AE = 2 CE$,
@@ -294,7 +294,7 @@ The parallel line to $AC$ of $E,F$ intersect $BC$ at $G,H$ respectively,
 Prove that $EG + FH = AC$-/
 
 -- We have triangle $\triangle ABC$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma A_ne_B : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
@@ -351,7 +351,7 @@ the angle bisector of $\angle ABC$ intersect $AD$ and $AC$ at $M,N$ respectively
 Prove that $AB^2 - AN^2 = BM \times BN$-/
 
 -- Let triangle $\triangle ABC$ be a right triangle with $\angle BAC = 90^{circ}$
-variable {A B C : P} {hnd : ¬ collinear A B C}
+variable {A B C : P} {hnd : ¬ Collinear A B C}
 -- Claim: $A \ne B$ and $B \ne C$ and $C \ne A$.
 lemma a_ne_b : A ≠ B := sorry
 lemma B_ne_C : B ≠ C := sorry
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1h'.lean b/EuclideanGeometry/Example/ShanZun/Ex1h'.lean
index df140326..c8da9eab 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1h'.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1h'.lean
@@ -107,15 +107,15 @@ structure Setting (Plane : Type _) [EuclideanPlane Plane] where
 --The pair of points (G,H) divides the segement AC into three segments with the same length.
 theorem result {Plane : Type _} [EuclideanPlane Plane] (e : Setting Plane) :(SEG e.A e.G).length=(SEG e.G e.H).length ∧ (SEG e.A e.G).length=(SEG e.H e.C).length := by
   -- $▵AFG$ is nontrivial
-  have AFG_nd : ¬ collinear e.A e.F e.G := sorry
+  have AFG_nd : ¬ Collinear e.A e.F e.G := sorry
   -- $▵CBG$ is nontrivial
-  have CBG_nd : ¬ collinear e.C e.B e.G :=sorry
+  have CBG_nd : ¬ Collinear e.C e.B e.G :=sorry
   -- $▵ AFG ∼ ▵ CBG$ with ratio 1/2 by AAA
   have AFG_sim_CBG : (TRI_nd e.A e.F e.G AFG_nd) ∼ (TRI_nd e.C e.B e.G CBG_nd) := sorry
 --The following three may not be necessary since we can use the above three lemmas instead
 
-  have CEH_nd : ¬ collinear e.C e.E e.H := sorry
-  have ADH_nd : ¬ collinear e.A e.D e.H := sorry
+  have CEH_nd : ¬ Collinear e.C e.E e.H := sorry
+  have ADH_nd : ¬ Collinear e.A e.D e.H := sorry
   -- $▵ CEH ∼ ▵ ADH$ with ratio 1/2 by AAA
   have CEH_sim_ADH : (TRI_nd e.C e.E e.H CEH_nd) ∼ (TRI_nd e.A e.D e.H ADH_nd) := sorry
   -- length of segment AG is half of segment CG
diff --git a/EuclideanGeometry/Example/ShanZun/Ex1h.lean b/EuclideanGeometry/Example/ShanZun/Ex1h.lean
index c3e864dd..b1f7bb12 100644
--- a/EuclideanGeometry/Example/ShanZun/Ex1h.lean
+++ b/EuclideanGeometry/Example/ShanZun/Ex1h.lean
@@ -64,15 +64,15 @@ lemma b_ne_f : B ≠ F := sorry
 -- G is the intersection point of AC and BF, H is the intersection point of AC and DE
 variable {G H : P} {hg : is_inx G (LIN A C c_ne_a.symm) (LIN B F b_ne_f)} {hh : is_inx H (LIN A C c_ne_a.symm) (LIN D E d_ne_e)}
 -- $▵AFG$ is nontrivial
-lemma afg_nd : ¬ collinear A F G := sorry
+lemma afg_nd : ¬ Collinear A F G := sorry
 -- $▵CBG$ is nontrivial
-lemma cbg_nd : ¬ collinear C B G :=sorry
+lemma cbg_nd : ¬ Collinear C B G :=sorry
 -- $▵ AFG ∼ ▵ CBG$ with ratio 1/2 by AAA
 lemma afg_sim_cbg : (TRI_nd A F G afg_nd) ∼ (TRI_nd C B G cbg_nd) := sorry
 --The following three may not be necessary since we can use the above three lemmas instead
 --
-lemma ceh_nd : ¬ collinear C E H := sorry
-lemma adh_nd : ¬ collinear A D H :=sorry
+lemma ceh_nd : ¬ Collinear C E H := sorry
+lemma adh_nd : ¬ Collinear A D H :=sorry
 -- $▵ CEH ∼ ▵ ADH$ with ratio 1/2 by AAA
 lemma ceh_sim_adh : (TRI_nd C E H ceh_nd) ∼ (TRI_nd A D H adh_nd) := sorry
 -- length of segment AG is half of segment CG
diff --git a/EuclideanGeometry/Foundation/Axiom/Basic/Class.lean b/EuclideanGeometry/Foundation/Axiom/Basic/Class.lean
index e8ba91fb..1cce6e08 100644
--- a/EuclideanGeometry/Foundation/Axiom/Basic/Class.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Basic/Class.lean
@@ -42,14 +42,6 @@ variable {P : Type*} {α β γ}
 
 section carrier
 
-abbrev PtNe [EuclideanPlane P] (A B : P) : Prop :=
-  Fact <| A ≠ B
-
-instance PtNe.symm [EuclideanPlane P] {A B : P} [h : PtNe A B] : PtNe B A := ⟨h.out.symm⟩
-
-@[simp]
-lemma pt_ne [EuclideanPlane P] {A B : P} [h : PtNe A B] : A ≠ B := @Fact.out _ h
-
 /-- The class of plane figures. We say `α` is a plane figure, if for every given Euclidean plane `P`, `α P` is a collection of specific figures on `P`, each equipped with a carrier set of type `Set P`. -/
 class Fig (α : Type*) (P : outParam <| Type*) where
   carrier : α → Set P
diff --git a/EuclideanGeometry/Foundation/Axiom/Basic/Plane.lean b/EuclideanGeometry/Foundation/Axiom/Basic/Plane.lean
index c33ca522..b64163dc 100644
--- a/EuclideanGeometry/Foundation/Axiom/Basic/Plane.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Basic/Plane.lean
@@ -33,6 +33,14 @@ class EuclideanPlane (P : Type _) extends MetricSpace P, NormedAddTorsor Vec P
 
 variable {P : Type _} [EuclideanPlane P]
 
+abbrev PtNe (A B : P) : Prop :=
+  Fact <| A ≠ B
+
+instance PtNe.symm [EuclideanPlane P] {A B : P} [h : PtNe A B] : PtNe B A := ⟨h.out.symm⟩
+
+@[simp]
+lemma pt_ne [EuclideanPlane P] {A B : P} [h : PtNe A B] : A ≠ B := @Fact.out _ h
+
 def Vec.mkPtPt (A B : P) : Vec := (B -ᵥ A)
 
 scoped notation "VEC" => Vec.mkPtPt
@@ -60,14 +68,18 @@ theorem neg_vec_norm_eq (A B : P) : ‖- VEC A B‖ = ‖VEC A B‖ := by
 theorem vec_norm_eq_rev (A B : P) : ‖VEC A B‖ = ‖VEC B A‖ := by
   rw [← neg_vec, neg_vec_norm_eq]
 
-theorem eq_iff_vec_eq_zero (A B : P) : B = A ↔ VEC A B = 0 := vsub_eq_zero_iff_eq.symm
+theorem eq_iff_vec_eq_zero {A B : P} : B = A ↔ VEC A B = 0 := vsub_eq_zero_iff_eq.symm
 
-theorem ne_iff_vec_ne_zero (A B : P) : B ≠ A ↔ VEC A B ≠ 0 := (eq_iff_vec_eq_zero A B).not
+theorem ne_iff_vec_ne_zero {A B : P} : B ≠ A ↔ VEC A B ≠ 0 := eq_iff_vec_eq_zero.not
 
 @[simp]
 theorem vec_add_vec (A B C : P) : VEC A B + VEC B C = VEC A C := by
   rw [add_comm, Vec.mkPtPt, Vec.mkPtPt, Vec.mkPtPt, vsub_add_vsub_cancel]
 
+theorem vec_vadd_right (A B : P) (v : Vec) : VEC A (v +ᵥ B) = v + VEC A B := vadd_vsub_assoc v B A
+
+theorem vec_vadd_left (A B : P) (v : Vec) : VEC (v +ᵥ A) B = VEC A B - v := vsub_vadd_eq_vsub_sub B A v
+
 @[simp]
 theorem vec_of_pt_vadd_pt_eq_vec (A : P) (v : Vec) : VEC A (v +ᵥ A) = v := vadd_vsub v A
 
@@ -88,10 +100,10 @@ theorem vec_sub_vec' (O A B: P) : VEC A O - VEC B O = VEC A B := by
 theorem pt_eq_pt_of_eq_smul_smul {O A B : P} {v : Vec} {tA tB : ℝ} (h : tA = tB) (ha : VEC O A = tA • v) (hb : VEC O B = tB • v) : A = B := by
   have hc : VEC A B = VEC O B - VEC O A := (vec_sub_vec O A B).symm
   rw [ha, hb, ← sub_smul, Iff.mpr sub_eq_zero h.symm, zero_smul] at hc
-  exact ((eq_iff_vec_eq_zero A B).2 hc).symm
+  exact (eq_iff_vec_eq_zero.2 hc).symm
 
 
-def VecND.mkPtPt (A B : P) (h : B ≠ A) : VecND := ⟨Vec.mkPtPt A B, (ne_iff_vec_ne_zero A B).mp h⟩
+def VecND.mkPtPt (A B : P) (h : B ≠ A) : VecND := ⟨Vec.mkPtPt A B, ne_iff_vec_ne_zero.mp h⟩
 
 @[inherit_doc VecND.mkPtPt]
 scoped syntax "VEC_nd" ws term:max ws term:max (ws term:max)? : term
@@ -117,4 +129,63 @@ def delabVecNDMkPtPt : Delab := do
 @[simp]
 lemma VecND.coe_mkPtPt (A B : P) [_h : Fact (B ≠ A)] : VEC_nd A B = VEC A B := rfl
 
+/- Definition of the wedge of three points.-/
+
+section wedge
+
+def wedge (A B C : P) : ℝ := Vec.det (VEC A B) (VEC A C)
+
+def oarea (A B C : P) : ℝ := wedge A B C / 2
+
+@[simp]
+theorem wedge_self₁₂ (A C : P) : wedge A A C = 0 := by
+  unfold wedge
+  simp
+
+@[simp]
+theorem wedge_self₂₃ (A C : P) : wedge C A A = 0 := by
+  unfold wedge
+  simp
+
+@[simp]
+theorem wedge_self₃₁ (A C : P) : wedge A C A = 0 := by
+  unfold wedge
+  simp
+
+theorem wedge213 (A B C : P) : wedge B A C = - wedge A B C := by
+  unfold wedge
+  rw [← neg_vec A B, ← vec_sub_vec A B C, map_sub]
+  simp only [map_neg, LinearMap.neg_apply, Vec.det_self, neg_zero, sub_zero]
+
+theorem wedge132 (A B C : P) : wedge A C B = - wedge A B C := by
+  unfold wedge
+  rw [Vec.det_skew]
+
+theorem wedge231 (A B C : P) : wedge C A B = wedge A B C := by
+  rw [wedge213, wedge132, neg_neg]
+
+theorem wedge312 (A B C : P) : wedge B C A = wedge A B C := by rw [wedge231, wedge231]
+
+theorem wedge321 (A B C : P) : wedge C B A = - wedge A B C := by rw [wedge213, wedge312]
+
+lemma wedge_def₁ (A B C : P) : wedge A B C = Vec.det (VEC A B) (VEC A C) := rfl
+
+lemma wedge_def₂ (A B C : P) : wedge A B C = Vec.det (VEC B C) (VEC B A) := by
+  rw [← wedge_def₁, wedge231]
+
+lemma wedge_def₃ (A B C : P) : wedge A B C = Vec.det (VEC C A) (VEC C B) := by
+  rw [← wedge_def₂, wedge231]
+
+lemma wedge_vadd (A B C : P) (v : Vec) : wedge A B (v +ᵥ C) = Vec.det (VEC A B) v + wedge A B C := by
+  simp [wedge, vec_vadd_right]
+
+lemma wedge_smul_vec_self_vadd (A B C : P) (k : ℝ) : wedge A B (k • VEC A B +ᵥ C) = wedge A B C := by
+  simp [wedge_vadd]
+
+lemma wedge_eq_wedge_iff (A B C D : P) [PtNe B A] : wedge A B C = wedge A B D ↔ ∃ k : ℝ, VEC C D = k • VEC A B := by
+  rw [eq_comm, ← sub_eq_zero, wedge, wedge, ← map_sub, vec_sub_vec, Vec.det_eq_zero_iff_eq_smul_left,
+    or_iff_right (ne_iff_vec_ne_zero.mp pt_ne)]
+
+end wedge
+
 end EuclidGeom
diff --git a/EuclideanGeometry/Foundation/Axiom/Circle/Basic.lean b/EuclideanGeometry/Foundation/Axiom/Circle/Basic.lean
index 507d6d09..0a2bce79 100644
--- a/EuclideanGeometry/Foundation/Axiom/Circle/Basic.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Circle/Basic.lean
@@ -23,7 +23,7 @@ def mk_pt_pt (O A : P) [h : PtNe O A] : Circle P where
   radius := dist O A
   rad_pos := dist_pos.mpr h.out
 
-def mk_pt_pt_pt (A B C: P) (h : ¬ collinear A B C) : Circle P := sorry
+def mk_pt_pt_pt (A B C: P) (h : ¬ Collinear A B C) : Circle P := sorry
 
 end Circle
 
@@ -156,7 +156,7 @@ end Circle
 end position
 
 
-section collinear
+section Collinear
 
 namespace Circle
 
@@ -182,8 +182,8 @@ lemma pts_lieson_circle_vec_eq {A B : P} {ω : Circle P} [hne : PtNe B A] (hl₁
     linarith⟩
   apply distinct_pts_same_dist_vec_eq
   · have : (perp_foot ω.center (LIN A B)) LiesOn (LIN A B) := perp_foot_lies_on_line _ _
-    have : collinear A B (perp_foot ω.center (LIN A B)) := Line.pt_pt_linear this
-    have : collinear (perp_foot ω.center (LIN A B)) A B := perm_collinear_trd_fst_snd this
+    have : Collinear A B (perp_foot ω.center (LIN A B)) := Line.pt_pt_linear this
+    have : Collinear (perp_foot ω.center (LIN A B)) A B := Collinear.perm₃₁₂ this
     apply Line.pt_pt_maximal this
   apply (sq_eq_sq dist_nonneg dist_nonneg).mp
   calc
@@ -201,12 +201,12 @@ theorem pts_lieson_circle_perpfoot_eq_midpoint {A B : P} {ω : Circle P} [hne :
       _ = (1 / 2 : ℝ) • ((2 : ℝ) • VEC A (perp_foot ω.center (LIN A B))) := by rw [← (pts_lieson_circle_vec_eq hl₁ hl₂), two_smul]
       _ = VEC A (perp_foot ω.center (LIN A B)) := by simp
   have eq₂ : VEC A (SEG A B).midpoint = (1 / 2 : ℝ) • (VEC A B) := Seg.vec_source_midpt
-  apply (eq_iff_vec_eq_zero _ _).mpr
+  apply eq_iff_vec_eq_zero.mpr
   calc
     _ = VEC A (perp_foot ω.center (LIN A B)) - VEC A (SEG A B).midpoint := by rw [vec_sub_vec]
     _ = 0 := by rw [eq₁, eq₂]; simp
 
-theorem three_pts_lieson_circle_not_collinear {A B C : P} {ω : Circle P} [hne₁ : PtNe B A] [hne₂ : PtNe C B] [hne₃ : PtNe A C] (hl₁ : A LiesOn ω) (hl₂ : B LiesOn ω) (hl₃ : C LiesOn ω) : ¬ (collinear A B C) := by
+theorem three_pts_lieson_circle_not_collinear {A B C : P} {ω : Circle P} [hne₁ : PtNe B A] [hne₂ : PtNe C B] [hne₃ : PtNe A C] (hl₁ : A LiesOn ω) (hl₂ : B LiesOn ω) (hl₃ : C LiesOn ω) : ¬ (Collinear A B C) := by
   intro h
   have eq₁ : VEC A (perp_foot ω.center (LIN A B)) = VEC (perp_foot ω.center (LIN A B)) B := pts_lieson_circle_vec_eq hl₁ hl₂
   have eq₂ : VEC A (perp_foot ω.center (LIN A C)) = VEC (perp_foot ω.center (LIN A C)) C := pts_lieson_circle_vec_eq hl₁ hl₃
@@ -218,12 +218,12 @@ theorem three_pts_lieson_circle_not_collinear {A B C : P} {ω : Circle P} [hne
     calc
       _ = VEC (perp_foot ω.center (LIN A B)) C - VEC (perp_foot ω.center (LIN A B)) B := by rw [vec_sub_vec]
       _ = 0 := by rw [← eq₁, ← eq₂, sub_self]
-  have : VEC B C ≠ 0 := (ne_iff_vec_ne_zero _ _).mp hne₂.out
+  have : VEC B C ≠ 0 := ne_iff_vec_ne_zero.mp hne₂.out
   tauto
 
 end Circle
 
-end collinear
+end Collinear
 
 section antipode
 
@@ -255,7 +255,7 @@ theorem antipode_distinct {A : P} {ω : Circle P} {ha : A LiesOn ω} : antipode
     apply (eq_vadd_iff_vsub_eq _ _ _).mp eq.symm
   have neq : A ≠ ω.center := (pt_lieson_ne_center ha).out
   contrapose! neq
-  apply (eq_iff_vec_eq_zero _ _).mpr
+  apply eq_iff_vec_eq_zero.mpr
   have : 2 • (VEC ω.center A) = 0 := by
     rw [two_smul]
     nth_rw 1 [this]
diff --git a/EuclideanGeometry/Foundation/Axiom/Circle/CCPosition.lean b/EuclideanGeometry/Foundation/Axiom/Circle/CCPosition.lean
index 700a92d8..0ff726c2 100644
--- a/EuclideanGeometry/Foundation/Axiom/Circle/CCPosition.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Circle/CCPosition.lean
@@ -106,7 +106,7 @@ theorem extangent_pt_lieson_circles {ω₁ : Circle P} {ω₂ : Circle P} (h : 
           rw [norm_smul, Dir.norm_unitVec, mul_one, Real.norm_of_nonneg ω₂.rad_pos.le]
   exact this
 
-theorem extangent_pt_centers_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Extangent ω₂) : collinear ω₁.center (Extangentpt h) ω₂.center := by
+theorem extangent_pt_centers_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Extangent ω₂) : Collinear ω₁.center (Extangentpt h) ω₂.center := by
   haveI : PtNe ω₁.center ω₂.center := ⟨extangent_centers_distinct h⟩
   have : VEC ω₁.center (Extangentpt h) = (ω₁.radius * ‖VEC ω₁.center ω₂.center‖⁻¹) • (VEC ω₁.center ω₂.center) := by
     calc
@@ -115,7 +115,7 @@ theorem extangent_pt_centers_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h
         simp
       _ = ω₁.radius • (‖VEC ω₁.center ω₂.center‖⁻¹ • (VEC ω₁.center ω₂.center)) := rfl
       _ = (ω₁.radius * ‖VEC ω₁.center ω₂.center‖⁻¹) • (VEC ω₁.center ω₂.center) := by apply smul_smul
-  apply flip_collinear_snd_trd (collinear_of_vec_eq_smul_vec this)
+  apply Collinear.perm₁₃₂ (collinear_of_vec_eq_smul_vec this)
 
 end CC
 
@@ -176,7 +176,7 @@ theorem intangent_pt_lieson_circles {ω₁ : Circle P} {ω₂ : Circle P} (h : 
   show Circle.IsOn (Intangentpt h) ω₂
   exact this
 
-theorem intangentpt_centers_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Intangent ω₂) : collinear ω₁.center ω₂.center (Intangentpt h) := by
+theorem intangentpt_centers_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Intangent ω₂) : Collinear ω₁.center ω₂.center (Intangentpt h) := by
   haveI : PtNe ω₁.center ω₂.center := ⟨h.2⟩
   have : VEC ω₁.center (Intangentpt h) = (- ω₁.radius * ‖(VEC ω₁.center ω₂.center)‖⁻¹) • VEC ω₁.center ω₂.center := by
     calc
@@ -298,7 +298,7 @@ def Inxpts {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Intersect ω₂) : C
   right := (- Real.sqrt (ω₁.radius ^ 2 - (radical_axis_dist_to_the_first ω₁ ω₂) ^ 2) * Complex.I + (radical_axis_dist_to_the_first ω₁ ω₂)) • (VEC_nd ω₁.center ω₂.center (intersected_centers_distinct h).symm).toDir.unitVec +ᵥ ω₁.center
 
 theorem inx_pts_distinct {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Intersect ω₂) : (Inxpts h).left ≠ (Inxpts h).right := by
-  apply (ne_iff_vec_ne_zero _ _).mpr
+  apply ne_iff_vec_ne_zero.mpr
   unfold Vec.mkPtPt Inxpts
   simp only [neg_mul, vadd_vsub_vadd_cancel_right, ne_eq, ← sub_smul]
   simp only [add_sub_add_right_eq_sub, sub_neg_eq_add, smul_eq_zero, add_self_eq_zero, mul_eq_zero,
@@ -391,11 +391,11 @@ theorem inx_pts_lieson_circles {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁
       apply Real.sqrt_sq (by linarith [ω₂.rad_pos])
       apply mul_ne_zero (by norm_num) (dist_ne_zero.mpr cne.out)
 
-lemma inx_pts_not_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Intersect ω₂) : (¬ collinear (Inxpts h).left ω₁.center ω₂.center) ∧ (¬ collinear (Inxpts h).right ω₁.center ω₂.center) := by
+lemma inx_pts_not_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Intersect ω₂) : (¬ Collinear (Inxpts h).left ω₁.center ω₂.center) ∧ (¬ Collinear (Inxpts h).right ω₁.center ω₂.center) := by
   constructor
   · intro hc
     set tri : Triangle P := ▵ (Inxpts h).left ω₁.center ω₂.center with tri_def
-    have : collinear tri.1 tri.2 tri.3 := hc
+    have : Collinear tri.1 tri.2 tri.3 := hc
     rw [Triangle.edge_sum_eq_edge_iff_collinear] at this
     rcases this with heq | (heq | heq)
     · rw [tri_def] at heq
@@ -430,7 +430,7 @@ lemma inx_pts_not_collinear {ω₁ : Circle P} {ω₂ : Circle P} (h : ω₁ Int
     linarith
   intro hc
   set tri : Triangle P := ▵ (Inxpts h).right ω₁.center ω₂.center with tri_def
-  have : collinear tri.1 tri.2 tri.3 := hc
+  have : Collinear tri.1 tri.2 tri.3 := hc
   rw [Triangle.edge_sum_eq_edge_iff_collinear] at this
   rcases this with heq | (heq | heq)
   · rw [tri_def] at heq
@@ -513,7 +513,7 @@ theorem inx_pts_line_perp_center_line {ω₁ : Circle P} {ω₂ : Circle P} (h :
         rw [← hn]
         show ‖VEC (Inxpts h).left (Inxpts h).right‖ ≠ 0
         apply norm_ne_zero_iff.mpr
-        apply (ne_iff_vec_ne_zero _ _).mp (inx_pts_distinct h).symm
+        apply ne_iff_vec_ne_zero.mp (inx_pts_distinct h).symm
   have hdir: (VEC_nd (Inxpts h).left (Inxpts h).right (inx_pts_distinct h).symm).toDir = - (Dir.I * (VEC_nd ω₁.center ω₂.center (intersected_centers_distinct h).symm).toDir) := by
     ext; rw [this]; rw [this]
   calc
diff --git a/EuclideanGeometry/Foundation/Axiom/Circle/IncribedAngle.lean b/EuclideanGeometry/Foundation/Axiom/Circle/IncribedAngle.lean
index 2e25fe40..8e83dc17 100644
--- a/EuclideanGeometry/Foundation/Axiom/Circle/IncribedAngle.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Circle/IncribedAngle.lean
@@ -91,8 +91,8 @@ def complement (β : Arc P) : Arc P where
 lemma liesint_arc_not_lieson_dlin {β : Arc P} {p : P} (h : p LiesInt β) : ¬ (p LiesOn (DLIN β.source β.target)) := by
   intro hl
   have hl : p LiesOn (LIN β.source β.target) := hl
-  have hco : collinear β.source β.target p := Line.pt_pt_linear hl
-  have hco' : ¬ (collinear β.source β.target p) := Circle.three_pts_lieson_circle_not_collinear (hne₂ := ⟨h.2.2⟩) (hne₃ := ⟨h.2.1.symm⟩) β.ison.1 β.ison.2 h.1.1
+  have hco : Collinear β.source β.target p := Line.pt_pt_linear hl
+  have hco' : ¬ (Collinear β.source β.target p) := Circle.three_pts_lieson_circle_not_collinear (hne₂ := ⟨h.2.2⟩) (hne₃ := ⟨h.2.1.symm⟩) β.ison.1 β.ison.2 h.1.1
   tauto
 
 theorem liesint_arc_liesonright_dlin {β : Arc P} {p : P} (h : p LiesInt β) : p LiesOnRight (DLIN β.source β.target) := by
@@ -132,7 +132,7 @@ theorem cangle_of_complementary_arc_are_opposite (β : Arc P) : β.cangle.value
   show (∠ β.source β.circle.center β.target = -∠ β.target β.circle.center β.source)
   apply neg_value_of_rev_ang
 
-theorem antipode_iff_collinear (A B : P) {ω : Circle P} [h : PtNe B A] (h₁ : A LiesOn ω) (h₂ : B LiesOn ω) : Arc.IsAntipode A B h₁ h₂ ↔ collinear ω.center A B := by
+theorem antipode_iff_collinear (A B : P) {ω : Circle P} [h : PtNe B A] (h₁ : A LiesOn ω) (h₂ : B LiesOn ω) : Arc.IsAntipode A B h₁ h₂ ↔ Collinear ω.center A B := by
   haveI : PtNe A ω.center := Circle.pt_lieson_ne_center h₁
   haveI : PtNe B ω.center := Circle.pt_lieson_ne_center h₂
   constructor
@@ -160,8 +160,8 @@ theorem antipode_iff_collinear (A B : P) {ω : Circle P} [h : PtNe B A] (h₁ :
   exact value_eq_pi_of_lies_int_seg_nd hli
 
 theorem mk_pt_pt_diam_isantipode {A B : P} [h : PtNe A B] : Arc.IsAntipode A B (Circle.mk_pt_pt_diam_fst_lieson) (Circle.mk_pt_pt_diam_snd_lieson) := by
-  have hc : collinear (SEG A B).midpoint A B := by
-    apply perm_collinear_trd_fst_snd
+  have hc : Collinear (SEG A B).midpoint A B := by
+    apply Collinear.perm₃₁₂
     apply Line.pt_pt_linear
     show (SEG A B).midpoint LiesOn (SEG_nd A B).toLine
     apply SegND.lies_on_toLine_of_lie_on
diff --git a/EuclideanGeometry/Foundation/Axiom/Circle/LCPosition.lean b/EuclideanGeometry/Foundation/Axiom/Circle/LCPosition.lean
index c5232b41..4520d488 100644
--- a/EuclideanGeometry/Foundation/Axiom/Circle/LCPosition.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Circle/LCPosition.lean
@@ -276,8 +276,8 @@ theorem DirLC_intersection_eq_inxpts {l : DirLine P} {ω : Circle P} {A : P} (h
     have : dist_pt_line ω.center l = ω.radius := h''
     have : dist_pt_line ω.center l < ω.radius := h'
     linarith⟩
-  have hc : collinear A (Inxpts h).front (Inxpts h).back := Line.linear h₁ (inx_pts_lieson_dlin h).1 (inx_pts_lieson_dlin h).2
-  have hnc : ¬ (collinear A (Inxpts h).front (Inxpts h).back) := three_pts_lieson_circle_not_collinear h₃ (inx_pts_lieson_circle h).1 (inx_pts_lieson_circle h).2
+  have hc : Collinear A (Inxpts h).front (Inxpts h).back := Line.linear h₁ (inx_pts_lieson_dlin h).1 (inx_pts_lieson_dlin h).2
+  have hnc : ¬ (Collinear A (Inxpts h).front (Inxpts h).back) := three_pts_lieson_circle_not_collinear h₃ (inx_pts_lieson_circle h).1 (inx_pts_lieson_circle h).2
   tauto
 
 theorem pt_pt_tangent_eq_tangent_pt {A B : P} {ω : Circle P} (h₁ : A LiesOut ω) (h₂ : B LiesOn ω) (ht : (DLIN A B (pt_liesout_ne_pt_lieson h₁ h₂).out.symm) Tangent ω) : B = Tangentpt ht := by
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Class.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Class.lean
index 46586fb5..873d60ac 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Class.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Class.lean
@@ -42,7 +42,7 @@ noncomputable section
 namespace EuclidGeom
 
 class LinFig (α : Type*) (P : outParam <| Type*) [outParam <| EuclideanPlane P] extends Fig α P where
-  collinear' : ∀ {A B C : P} {F : α}, A LiesOn F → B LiesOn F → C LiesOn F → collinear A B C
+  collinear' : ∀ {A B C : P} {F : α}, A LiesOn F → B LiesOn F → C LiesOn F → Collinear A B C
 
 class ProjObj (β : Type _) where
   toProj : β → Proj
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Collinear.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Collinear.lean
index fccfad1c..8153b7c4 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Collinear.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Collinear.lean
@@ -10,164 +10,134 @@ open Classical
 
 variable {P : Type _} [EuclideanPlane P]
 
-section collinear
-
-/-- Given three distinct (ordered) points $A$, $B$, $C$, this function returns whether they are collinear, i.e. whether the projective direction of the vector $\overrightarrow{AB}$ is the same as the projective direction of the vector $\overrightarrow{AC}$. -/
-def collinear_of_nd {A B C : P} [_hac : PtNe A C] [_hba : PtNe B A] : Prop :=
-  VecND.toProj (VEC_nd A B) = VecND.toProj (VEC_nd A C)
+section Collinear
 
 /-- Given three points $A$, $B$, $C$, return whether they are collinear: if at least two of them are equal, then they are considered collinear; if the three points are distinct, we use the earlier definition of colinarity for distinct points. -/
-def collinear (A B C : P) : Prop :=
-  if h : (C = B) ∨ (A = C) ∨ (B = A) then True
-  else by
-    push_neg at h
-    exact collinear_of_nd (_hac := ⟨h.2.1⟩) (_hba := ⟨h.2.2⟩)
+def Collinear (A B C : P) : Prop := wedge A B C = 0
+
+theorem collinear_iff_wedge_eq_zero (A B C : P) : (Collinear A B C) ↔ (wedge A B C = 0) := .rfl
 
--- The definition of collinear now includes two cases: the degenerate case and the nondegenerate case. We use to_dir' to avoid problems involving using conditions of an "if" in its "then" and "else". And we only use VEC to define collinear.
+theorem not_collinear_iff_wedge_ne_zero (A B C : P) : (¬ Collinear A B C) ↔ (wedge A B C ≠ 0) := .rfl
+
+lemma collinear_iff_collinear {A B C D : P} [PtNe B A] (h : ∃ k : ℝ, VEC C D = k • VEC A B) : Collinear A B C ↔ Collinear A B D := by
+  unfold Collinear
+  rw [← wedge_eq_wedge_iff] at h
+  rw [h]
 
 /-- Given three points $A$, $B$, $C$ and a real number $t$, if the vector $\overrightarrow{AC}$ is $t$ times the vector $\overrightarrow{AB}$, then $A$, $B$, and $C$ are collinear. -/
-theorem collinear_of_vec_eq_smul_vec {A B C : P} {t : ℝ} (e : VEC A C = t • VEC A B) : collinear A B C := by
-  have : collinear A B C = True := by
-    unfold collinear
-    apply dite_eq_left_iff.mpr
-    simp only [eq_iff_iff, iff_true]
-    intro h'
-    push_neg at h'
-    unfold collinear_of_nd
-    rw [eq_comm, VecND.toProj_eq_toProj_iff]
-    exact ⟨t, e⟩
-  tauto
+theorem collinear_of_vec_eq_smul_vec {A B C : P} {t : ℝ} (e : VEC A C = t • VEC A B) : Collinear A B C := by
+  unfold Collinear wedge
+  rw [e]
+  simp
 
 /-- Given three points $A$, $B$, $C$, if the vector $\overrightarrow{AC}$ is a scalar multiple of the vector $\overrightarrow{AB}$, then $A$, $B$, $C$ are collinear. -/
-theorem collinear_of_vec_eq_smul_vec' {A B C : P} : (∃ t : ℝ, VEC A C = t • VEC A B) → collinear A B C := by
+theorem collinear_of_vec_eq_smul_vec' {A B C : P} : (∃ t : ℝ, VEC A C = t • VEC A B) → Collinear A B C := by
   intro ⟨_, e⟩
   exact collinear_of_vec_eq_smul_vec e
 
 /-- Given three points $A$, $B$, $C$ such that $B \neq A$, we have $A$, $B$, $C$ are collinear if and only if the vector $\overrightarrow{AC}$ is a scalar multiple of the vector $\overrightarrow{AB}$. -/
-theorem collinear_iff_eq_smul_vec_of_ne {A B C : P} [g : PtNe B A] : collinear A B C ↔ ∃ r : ℝ , VEC A C = r • VEC A B := by
+theorem collinear_iff_eq_smul_vec_of_ne {A B C : P} [g : PtNe B A] : Collinear A B C ↔ ∃ r : ℝ, VEC A C = r • VEC A B := by
   constructor
   · intro c
-    rw [← iff_true (collinear A B C), ← eq_iff_iff] at c
-    unfold collinear at c
-    rw [dite_eq_left_iff] at c
-    by_cases h : (C = B) ∨ (A = C) ∨ (B = A)
-    · by_cases h : C = A
-      · use 0
-        rw [h]
-        simp only [vec_same_eq_zero, zero_smul]
-      · have h : B = C := by have := g.out; tauto
-        use 1
-        rw [h]
-        simp only [one_smul]
-    · specialize c h
-      push_neg at h
-      unfold collinear_of_nd at c
-      simp only [ne_eq, eq_iff_iff, iff_true] at c
-      rwa [eq_comm, VecND.toProj_eq_toProj_iff] at c
-  · intro ⟨_, e⟩
-    exact collinear_of_vec_eq_smul_vec e
-
+    unfold Collinear wedge at c
+    rwa [Vec.det_eq_zero_iff_eq_smul_left, or_iff_right] at c
+    rw [← Ne, ← ne_iff_vec_ne_zero]
+    exact g.elim
+  · exact collinear_of_vec_eq_smul_vec'
 
 -- Please rewrite this part, use minimal theorems, but create a tactic called `collinearity`
 /-- For any two points $A$ and $C$, the points $A$, $A$, $C$ are collinear. -/
-theorem triv_collinear (A C : P) : (collinear A A C) := by
-  rw [← iff_true (collinear A A C), ← eq_iff_iff]
-  unfold collinear
-  rw [dite_eq_left_iff]
-  intro h
-  push_neg at h
-  exfalso
-  exact h.2.2 rfl
+@[simp]
+theorem triv_collinear₁₂ (A C : P) : (Collinear A A C) := by
+  unfold Collinear
+  simp
 
-theorem collinear_of_trd_eq_snd (A : P) {B C : P} (h : C = B) : collinear A B C :=
-  (dite_prop_iff_or (C = B ∨ A = C ∨ B = A)).mpr (.inl ⟨.inl h, trivial⟩)
+/-- For any two points $A$ and $C$, the points $A$, $C$, $C$ are collinear. -/
+@[simp]
+theorem triv_collinear₁₂₂₃ (A C : P) : (Collinear A C C) := by
+  unfold Collinear
+  simp
 
-theorem collinear_of_fst_eq_snd (B : P) {A C : P} (h : A = C) : collinear A B C :=
-  (dite_prop_iff_or (C = B ∨ A = C ∨ B = A)).mpr (.inl ⟨.inr (.inl h), trivial⟩)
+/-- For any two points $A$ and $C$, the points $A$, $C$, $A$ are collinear. -/
+@[simp]
+theorem triv_collinear₁₂₃₁ (A C : P) : (Collinear A C A) := by
+  unfold Collinear
+  simp
 
-theorem collinear_of_snd_eq_fst {A B : P} (C : P) (h : B = A)  : collinear A B C :=
-  (dite_prop_iff_or (C = B ∨ A = C ∨ B = A)).mpr (.inl ⟨.inr (.inr h), trivial⟩)
+theorem collinear_of_trd_eq_snd (A : P) {B C : P} (h : C = B) : Collinear A B C := by
+  simp [h]
+
+theorem collinear_of_fst_eq_snd (B : P) {A C : P} (h : A = C) : Collinear A B C := by
+  simp [h]
+
+theorem collinear_of_snd_eq_fst {A B : P} (C : P) (h : B = A)  : Collinear A B C := by
+  simp [h]
 
 /-- Given three points $A$, $B$, and $C$, if $A$, $B$, $C$ are collinear (in that order), then $A$, $C$, $B$ are collinear (in that order); in other words, swapping the last two of the three points does not change the definition of colinarity. -/
-theorem flip_collinear_snd_trd {A B C : P} (c : collinear A B C) : collinear A C B := by
-  by_cases h : B ≠ A ∧ C ≠ A
-  · haveI : PtNe B A := ⟨h.1⟩
-    rcases collinear_iff_eq_smul_vec_of_ne.1 c with ⟨t, e⟩
-    have ht : t ≠ 0 := by
-      by_contra ht'
-      rw [ht', zero_smul] at e
-      have _ : C = A := ((eq_iff_vec_eq_zero A C).2 e)
-      tauto
-    exact collinear_of_vec_eq_smul_vec (Eq.symm ((inv_smul_eq_iff₀ ht).2 e))
-  · rw [← iff_true (collinear _ _ _), ← eq_iff_iff]
-    unfold collinear
-    rw [dite_eq_left_iff]
-    intro g
-    exfalso
-    push_neg at *
-    exact g.2.2 $ h g.2.1.symm
+theorem collinear132 {A B C : P} : Collinear A B C ↔ Collinear A C B := by
+  unfold Collinear
+  rw [wedge132, neg_eq_zero]
+
+alias ⟨Collinear.perm₁₃₂, _⟩ := collinear132
 
 /-- Given three points $A$, $B$, and $C$, if $A$, $B$, $C$ are collinear (in that order), then $B$, $A$, $C$ are collinear (in that order); in other words, in the definition of colinarity, swapping the first two of the three points does not change property of the three points being collinear. -/
-theorem flip_collinear_fst_snd {A B C : P} (c : collinear A B C) : collinear B A C := by
-  by_cases h : B = A
-  · rw [h]
-    exact triv_collinear _ _
-  · haveI : PtNe B A := ⟨h⟩
-    rw [collinear_iff_eq_smul_vec_of_ne] at c
-    rcases c with ⟨r, e⟩
-    have e' : VEC B C = (1 - r) • VEC B A := by
-      rw [← vec_sub_vec A B C, e, ← neg_vec A B, smul_neg, sub_smul, neg_sub, one_smul]
-    exact collinear_of_vec_eq_smul_vec e'
-
-theorem perm_collinear_snd_trd_fst {A B C : P} (h : collinear A B C) : collinear B C A :=
-  flip_collinear_snd_trd (flip_collinear_fst_snd h)
-
-theorem perm_collinear_trd_fst_snd {A B C : P} (h : collinear A B C) : collinear C A B :=
-  perm_collinear_snd_trd_fst (perm_collinear_snd_trd_fst h)
-
-theorem flip_collinear_fst_trd {A B C : P} (h : collinear A B C) : collinear C B A :=
-  perm_collinear_snd_trd_fst (flip_collinear_snd_trd h)
+theorem collinear213 {A B C : P} : Collinear A B C ↔ Collinear B A C := by
+  unfold Collinear
+  rw [wedge213, neg_eq_zero]
+
+alias ⟨Collinear.perm₂₁₃, _⟩ := collinear213
+
+theorem collinear231 {A B C : P} : Collinear A B C ↔ Collinear B C A := by
+  unfold Collinear
+  rw [wedge231]
+
+alias ⟨Collinear.perm₂₃₁, _⟩ := collinear231
+
+theorem collinear312 {A B C : P} : Collinear A B C ↔ Collinear C A B := by
+  unfold Collinear
+  rw [wedge312]
+
+alias ⟨Collinear.perm₃₁₂, _⟩ := collinear312
+
+theorem collinear321 {A B C : P} : Collinear A B C ↔ Collinear C B A := by
+  unfold Collinear
+  rw [wedge321, neg_eq_zero]
+
+alias ⟨Collinear.perm₃₂₁, _⟩ := collinear321
 
 -- the proof of this theorem using def of line seems to be easier
 /-- Given four points $A$, $B$, $C$, $D$ with $B \neq A$, if $A$, $B$, $C$ are collinear, and if $A$, $B$, $D$ are collinear, then $A$, $C$, $D$ are collinear. -/
-theorem collinear_of_collinear_collinear_ne {A B C D: P} (h₁ : collinear A B C) (h₂ : collinear A B D) [h : PtNe B A] : (collinear A C D) := by
+theorem collinear_of_collinear_collinear_ne {A B C D: P} (h₁ : Collinear A B C) (h₂ : Collinear A B D) [h : PtNe B A] : (Collinear A C D) := by
   have ac : ∃ r : ℝ , VEC A C = r • VEC A B := collinear_iff_eq_smul_vec_of_ne.mp h₁
   have ad : ∃ s : ℝ , VEC A D = s • VEC A B := collinear_iff_eq_smul_vec_of_ne.mp h₂
   rcases ac with ⟨r,eq⟩
   rcases ad with ⟨s,eq'⟩
   by_cases nd : r = 0
   . simp only [nd, zero_smul] at eq
-    have : C = A := (eq_iff_vec_eq_zero A C).mpr eq
-    rw [this] ; apply triv_collinear
+    have : C = A := eq_iff_vec_eq_zero.mpr eq
+    rw [this] ; apply triv_collinear₁₂
   apply collinear_of_vec_eq_smul_vec'
   rw [eq,eq']
   use s/r
   rw [smul_smul, div_mul_cancel _ nd]
 
+set_option push_neg.use_distrib true in
 /-- Given three points $A$, $B$, $C$, if they are not collinear, then they are pairwise distinct, i.e. $C \neq B$, $A \neq C$, and $B \neq A$. -/
-theorem ne_of_not_collinear {A B C : P} (h : ¬ collinear A B C) : (C ≠ B) ∧ (A ≠ C) ∧ (B ≠ A) := by
-  rw [← iff_true (collinear A B C), ← eq_iff_iff] at h
-  unfold collinear at h
-  rw [dite_eq_left_iff] at h
-  push_neg at h
-  rcases h with ⟨g, _⟩
-  tauto
-
-theorem collinear_iff_toProj_eq_of_ptNe {A B C : P} [hba : PtNe B A] [hca : PtNe C A] : collinear A B C ↔ (VEC_nd A B).toProj = (VEC_nd A C).toProj := by
-  if hbc : B = C then simp only [hbc, collinear_of_trd_eq_snd A rfl]
-  else
-    refine' ⟨fun h ↦ _, fun h ↦ _⟩
-    exact ((dite_prop_iff_and _).mp h).2 <| by
-      push_neg
-      exact ⟨Ne.symm hbc, hca.1.symm, hba.1⟩
-    exact (dite_prop_iff_and _).mpr ⟨fun _ ↦ trivial, fun _ ↦ h⟩
-
-end collinear
+theorem ne_of_not_collinear {A B C : P} (h : ¬ Collinear A B C) : (C ≠ B) ∧ (A ≠ C) ∧ (B ≠ A) := by
+  push_neg
+  contrapose! h
+  obtain (rfl | rfl | rfl) := h <;> simp
+
+theorem collinear_iff_toProj_eq_of_ptNe {A B C : P} [hba : PtNe B A] [hca : PtNe C A] : Collinear A B C ↔ (VEC_nd A C).toProj = (VEC_nd A B).toProj := by
+  rw [collinear_iff_eq_smul_vec_of_ne, VecND.toProj_eq_toProj_iff]
+  rfl
+
+end Collinear
 
 section compatibility
 
 /-- If $A$, $B$, $C$ are three points which lie on a ray, then they are collinear. -/
-theorem Ray.collinear_of_lies_on {A B C : P} {ray : Ray P} (hA : A LiesOn ray) (hB : B LiesOn ray) (hC : C LiesOn ray) : collinear A B C := by
+theorem Ray.collinear_of_lies_on {A B C : P} {ray : Ray P} (hA : A LiesOn ray) (hB : B LiesOn ray) (hC : C LiesOn ray) : Collinear A B C := by
   rcases hA with ⟨a,_,Ap⟩
   rcases hB with ⟨b,_,Bp⟩
   rcases hC with ⟨c,_,Cp⟩
@@ -181,22 +151,22 @@ theorem Ray.collinear_of_lies_on {A B C : P} {ray : Ray P} (hA : A LiesOn ray) (
       apply add_right_cancel_iff.mp nd
     rw [this] at ab
     rw [sub_self, zero_smul] at ab
-    have : B = A := by apply (eq_iff_vec_eq_zero A B).mpr ab
-    rw [this] ; apply triv_collinear
+    have : B = A := by apply eq_iff_vec_eq_zero.mpr ab
+    rw [this] ; apply triv_collinear₁₂
   apply collinear_of_vec_eq_smul_vec'
   use (c - a)/(b - a)
   rw [ac, ab, smul_smul, div_mul_cancel _ nd]
 
 /-- If $A$, $B$, $C$ are three points which lie on a segment, then they are collinear. -/
-theorem Seg.collinear_of_lies_on {A B C : P} {seg : Seg P} (hA : A LiesOn seg) (hB : B LiesOn seg) (hC : C LiesOn seg) : collinear A B C := by
+theorem Seg.collinear_of_lies_on {A B C : P} {seg : Seg P} (hA : A LiesOn seg) (hB : B LiesOn seg) (hC : C LiesOn seg) : Collinear A B C := by
   by_cases nd : seg.source =seg.target
   . rcases hA with ⟨_,_,_,a⟩
     simp only [nd, vec_same_eq_zero, smul_zero] at a
-    have a_d : A = seg.target := by apply (eq_iff_vec_eq_zero seg.target A).mpr a
+    have a_d : A = seg.target := by apply eq_iff_vec_eq_zero.mpr a
     rcases hB with ⟨_,_,_,b⟩
     simp only [nd, vec_same_eq_zero, smul_zero] at b
-    have b_d : B = seg.target := by apply (eq_iff_vec_eq_zero seg.target B).mpr b
-    rw [a_d,b_d] ; apply triv_collinear
+    have b_d : B = seg.target := by apply eq_iff_vec_eq_zero.mpr b
+    rw [a_d,b_d] ; apply triv_collinear₁₂
   rw [<-ne_eq] at nd
   let seg_nd := SegND.mk seg.source seg.target nd.symm
   have ha : A LiesOn seg_nd.1 := by apply hA
@@ -214,7 +184,7 @@ Note that we do not need all reverse, extension line,... here. instead we should
 end compatibility
 
 /-- There exists three points $A$, $B$, $C$ on the plane such that they are not collinear. -/
-theorem nontriv_of_plane {H : Type _} [h : EuclideanPlane H] : ∃ A B C : H, ¬(collinear A B C) := by
+theorem nontriv_of_plane {H : Type _} [h : EuclideanPlane H] : ∃ A B C : H, ¬(Collinear A B C) := by
   rcases h.nonempty with ⟨A⟩
   let B := (⟨1, 0⟩ : Vec) +ᵥ A
   let C := (⟨0, 1⟩ : Vec) +ᵥ A
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Line.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Line.lean
index 2f8893c5..76dd8ba9 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Line.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Line.lean
@@ -893,21 +893,23 @@ end lieson
 
 
 
-section collinear
+section Collinear
 
 namespace Line
 
 /-- Given three points $A$, $B$ and $C$, if $A$ and $B$ are distinct and $C$ lies on the line $AB$, then $A$, $B$, and $C$ are collinear. -/
-theorem pt_pt_linear {A B C : P} [_h : PtNe B A] (hc : C LiesOn (LIN A B)) : collinear A B C :=
+theorem pt_pt_linear {A B C : P} [_h : PtNe B A] (hc : C LiesOn (LIN A B)) : Collinear A B C :=
   if hcb : C = B then collinear_of_trd_eq_snd A hcb
   else if hac : A = C then collinear_of_fst_eq_snd B hac
-  else haveI : PtNe C B := ⟨hcb⟩
-  perm_collinear_trd_fst_snd <| (dite_prop_iff_or _).mpr <| .inr ⟨by push_neg; exact ⟨hac, Fact.out, hcb⟩,
-    ((lies_on_iff_eq_toProj_of_lies_on snd_pt_lies_on_mk_pt_pt).mp hc).trans <|
-      congrArg toProj line_of_pt_pt_eq_rev⟩
+  else haveI : PtNe C B := ⟨hcb⟩; haveI : PtNe A C := ⟨hac⟩; Collinear.perm₂₁₃ <| by
+    rw [Line.lies_on_iff_eq_toProj_of_lies_on snd_pt_lies_on_mk_pt_pt] at hc
+    rw [collinear_iff_toProj_eq_of_ptNe]
+    trans (VEC_nd A B).toProj
+    · exact hc
+    · exact Ray.toProj_eq_toProj_of_mk_pt_pt -- TODO: refactor
 
 /-- The theorem states that if three points $A$, $B$, and $C$ lie on the same line $l$, then they are collinear. -/
-theorem linear {l : Line P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) (h₃ : C LiesOn l) : collinear A B C := by
+theorem linear {l : Line P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) (h₃ : C LiesOn l) : Collinear A B C := by
   if h : B = A then exact collinear_of_snd_eq_fst C h
   else
   haveI : PtNe B A := ⟨h⟩
@@ -916,33 +918,33 @@ theorem linear {l : Line P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l)
   exact h₃
 
 /-- If $A$ and $B$ are two distinct points and $C$ is a point such that $A$, $B$ and $C$ are collinear, then $C$ lies on the line $AB$. -/
-theorem pt_pt_maximal {A B C : P} [_h : PtNe B A] (Co : collinear A B C) : C LiesOn (LIN A B) :=
+theorem pt_pt_maximal {A B C : P} [_h : PtNe B A] (Co : Collinear A B C) : C LiesOn (LIN A B) :=
   if hcb : C = B then by
     rw [hcb]
     exact snd_pt_lies_on_mk_pt_pt
   else haveI : PtNe C B := ⟨hcb⟩
   (lies_on_iff_eq_toProj_of_lies_on snd_pt_lies_on_mk_pt_pt).mpr <|
-    (collinear_iff_toProj_eq_of_ptNe.mp (perm_collinear_snd_trd_fst Co)).trans <|
+    (collinear_iff_toProj_eq_of_ptNe.mp Co.perm₂₁₃).trans <|
       congrArg Line.toProj (line_of_pt_pt_eq_rev (_h := _h)).symm
 
 /-- Given two distinct points $A$ and $B$ on a line $l$, if a point $C$ is so that $A$, $B$, and $C$ are collinear, then $C$ lines on $l$. -/
-theorem maximal {l : Line P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) [_h : PtNe B A] (Co : collinear A B C) : C LiesOn l := by
+theorem maximal {l : Line P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) [_h : PtNe B A] (Co : Collinear A B C) : C LiesOn l := by
   rw [← eq_line_of_pt_pt_of_ne h₁ h₂]
   exact pt_pt_maximal Co
 
 /-- Given two distinct points $A$ and $B$, a point $X$ lies on the line through $A$ and $B$ if and only if $A$, $B$, and $X$ are collinear. -/
-theorem lies_on_line_of_pt_pt_iff_collinear {A B : P} [_h : PtNe B A] (X : P) : (X LiesOn (LIN A B)) ↔ collinear A B X := ⟨
+theorem lies_on_line_of_pt_pt_iff_collinear {A B : P} [_h : PtNe B A] (X : P) : (X LiesOn (LIN A B)) ↔ Collinear A B X := ⟨
   fun hx ↦ (LIN A B).linear fst_pt_lies_on_mk_pt_pt snd_pt_lies_on_mk_pt_pt hx,
   fun c ↦ (LIN A B).maximal fst_pt_lies_on_mk_pt_pt snd_pt_lies_on_mk_pt_pt c⟩
 
 -- This is also a typical proof that shows how to use linear, maximal, nontriv of a line. Please write it shorter in future.
 
 /-- This theorem states that if $A$ and $B$ are two distinct points on a line $l$, then a point $C$ lies on $l$ if and only if $A$, $B$, and $C$ are collinear. -/
-theorem lies_on_iff_collinear_of_ne_lies_on_lies_on {A B : P} {l : Line P} [_h : PtNe B A] (ha : A LiesOn l) (hb : B LiesOn l) (C : P) : (C LiesOn l) ↔ collinear A B C :=
+theorem lies_on_iff_collinear_of_ne_lies_on_lies_on {A B : P} {l : Line P} [_h : PtNe B A] (ha : A LiesOn l) (hb : B LiesOn l) (C : P) : (C LiesOn l) ↔ Collinear A B C :=
   ⟨fun hc ↦ l.linear ha hb hc, fun c ↦ l.maximal ha hb c⟩
 
 /-- The given theorem is an equivalence statement between the collinearity of three points and the existence of a line on which all three points lie. -/
-theorem collinear_iff_exist_line_lies_on (A B C : P) : collinear A B C ↔ ∃ l : Line P, (A LiesOn l) ∧ (B LiesOn l) ∧ (C LiesOn l) := by
+theorem collinear_iff_exist_line_lies_on (A B C : P) : Collinear A B C ↔ ∃ l : Line P, (A LiesOn l) ∧ (B LiesOn l) ∧ (C LiesOn l) := by
   constructor
   · intro c
     by_cases h : PtNe B A
@@ -959,39 +961,39 @@ theorem collinear_iff_exist_line_lies_on (A B C : P) : collinear A B C ↔ ∃ l
     if h : PtNe B A then exact (lies_on_iff_collinear_of_ne_lies_on_lies_on ha hb C).mp hc
     else
       simp [PtNe, fact_iff] at h
-      simp only [h, collinear, or_true, dite_true]
+      simp [h]
 
 end Line
 
 namespace DirLine
 
 /-- If $A$, $B$ and $C$ are three points on the same directed line $l$, then they are collinear. -/
-theorem linear {l : DirLine P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) (h₃ : C LiesOn l) : collinear A B C :=
+theorem linear {l : DirLine P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) (h₃ : C LiesOn l) : Collinear A B C :=
   Line.linear h₁ h₂ h₃
 
 /-- If $A$, $B$ and $C$ are three collinear points in which $A \neq B$, then $C$ lies on the directed line associated to $AB$. -/
-theorem pt_pt_maximal {A B C : P} [_h : PtNe B A] (Co : collinear A B C) : C LiesOn (DLIN A B) :=
+theorem pt_pt_maximal {A B C : P} [_h : PtNe B A] (Co : Collinear A B C) : C LiesOn (DLIN A B) :=
   Line.pt_pt_maximal Co
 
 /-- Given two points $A$ and $B$ on a directed line $l$, if a point $C$ is so that $A$, $B$, and $C$ are collinear, then $C$ lies on $l$. -/
-theorem maximal {l : DirLine P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) [_h : PtNe B A] (Co : collinear A B C) : C LiesOn l :=
+theorem maximal {l : DirLine P} {A B C : P} (h₁ : A LiesOn l) (h₂ : B LiesOn l) [_h : PtNe B A] (Co : Collinear A B C) : C LiesOn l :=
   Line.maximal h₁ h₂ Co
 
 /-- Given two distinct points $A$ and $B$, a point $X$ lies on the directed line from $A$ to $B$ if and only if $A$, $B$, and $X$ are collinear. -/
-theorem lies_on_dirline_of_pt_pt_iff_collinear {A B : P} [_h : PtNe B A] (X : P) : (X LiesOn (DLIN A B)) ↔ collinear A B X :=
+theorem lies_on_dirline_of_pt_pt_iff_collinear {A B : P} [_h : PtNe B A] (X : P) : (X LiesOn (DLIN A B)) ↔ Collinear A B X :=
   Line.lies_on_line_of_pt_pt_iff_collinear X
 
 /-- Let $A$ and $B$ be two distinct points on a directed line $l$, then a point $C$ lies on $l$ if and only if $A$, $B$ and $C$ are collinear. -/
-theorem lies_on_iff_collinear_of_ne_lies_on_lies_on {A B : P} {l : DirLine P} [_h : PtNe B A] (ha : A LiesOn l) (hb : B LiesOn l) (C : P) : (C LiesOn l) ↔ collinear A B C :=
+theorem lies_on_iff_collinear_of_ne_lies_on_lies_on {A B : P} {l : DirLine P} [_h : PtNe B A] (ha : A LiesOn l) (hb : B LiesOn l) (C : P) : (C LiesOn l) ↔ Collinear A B C :=
   Line.lies_on_iff_collinear_of_ne_lies_on_lies_on ha hb C
 
 /-- The theorem states that three points $A$, $B$, and $C$ are collinear if and only if there exists a line $l$ such that $A$, $B$, and $C$ all lie on $l$. -/
-theorem collinear_iff_exist_line_lies_on (A B C : P) : collinear A B C ↔ ∃ l : Line P, (A LiesOn l) ∧ (B LiesOn l) ∧ (C LiesOn l) :=
+theorem collinear_iff_exist_line_lies_on (A B C : P) : Collinear A B C ↔ ∃ l : Line P, (A LiesOn l) ∧ (B LiesOn l) ∧ (C LiesOn l) :=
   Line.collinear_iff_exist_line_lies_on A B C
 
 end DirLine
 
-end collinear
+end Collinear
 
 
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Line_trash.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Line_trash.lean
index 55ed99f5..517c95c4 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Line_trash.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Line_trash.lean
@@ -9,7 +9,7 @@ theorem same_dist_eq_or_eq_neg {A B C : P} [hne : PtNe B A] (h : C LiesOn (LIN A
   have : LIN A B = (RAY A B).toLine := rfl
   rw [this] at h
   rcases Ray.lies_on_toLine_iff_lies_on_or_lies_on_rev.mp h with h | h
-  · left; apply (eq_iff_vec_eq_zero _ _).mpr
+  · left; apply eq_iff_vec_eq_zero.mpr
     have : VEC A C = (dist A C) • (RAY A B).2.unitVec := Ray.vec_eq_dist_smul_toDir_of_lies_on h
     calc
       _ = VEC A C - VEC A B := by rw [vec_sub_vec]
@@ -49,7 +49,7 @@ theorem midpoint_dist_eq_iff_eq_endpts {A B C : P} [hne : PtNe B C] (h : A LiesO
     rcases same_dist_eq_or_eq_neg this hh with h₁ | h₂
     · left; exact h₁
     right
-    apply (eq_iff_vec_eq_zero _ _).mpr
+    apply eq_iff_vec_eq_zero.mpr
     calc
       _ = VEC (SEG B C).midpoint A - VEC (SEG B C).midpoint C := by rw [vec_sub_vec]
       _ = VEC B (SEG B C).midpoint - VEC (SEG B C).midpoint C := by rw [h₂]
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Parallel.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Parallel.lean
index e150f9b8..76d29bcf 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Parallel.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Parallel.lean
@@ -46,21 +46,32 @@ instance : LinearObj (Line P) where
 
 -- Our definition of parallel for LinearObj is very general. Not only can it apply to different types of Objs, but also include degenerate cases, such as ⊆(inclusions), =(equal).
 /-- Given two linear objects $l_1$ and $l_2$ (not necessarily of the same type), this function returns whether they are parallel, defined as having the same associated projective direction. -/
-def parallel {α β : Type*} [ProjObj α] [ProjObj β] (l₁ : α) (l₂ : β) : Prop :=
+def Parallel {α β : Type*} [ProjObj α] [ProjObj β] (l₁ : α) (l₂ : β) : Prop :=
   toProj l₁ = toProj l₂
 
+/-- This is to rewrite \verb|parallel l l'| as \verb|l ParallelTo l'| -/
+scoped infix : 50 " ParallelTo " => Parallel
+
+/-- This is to rewrite \verb|parallel l l'| as $l \parallel l'$. -/
+scoped infix : 50 " ∥ " => Parallel
+
+@[refl]
+lemma Parallel.refl {α : Type*} [ProjObj α] (l : α) : l ∥ l := rfl
+
+@[symm]
+lemma Parallel.symm {α β : Type*} [ProjObj α] [ProjObj β] {l₁ : α} {l₂ : β} : l₁ ∥ l₂ → l₂ ∥ l₁ := Eq.symm
+
+lemma parallel_comm {α β : Type*} [ProjObj α] [ProjObj β] {l₁ : α} {l₂ : β} : l₁ ∥ l₂ ↔ l₂ ∥ l₁ := eq_comm
+
+@[trans]
+lemma Parallel.trans {α β γ : Type*} [ProjObj α] [ProjObj β] [ProjObj γ] {l₁ : α} {l₂ : β} {l₃ : γ} : l₁ ∥ l₂ → l₂ ∥ l₃ → l₁ ∥ l₃ := Eq.trans
+
 /-- Being parallel defines an equivalence relation among all linear objects, that is they satisfy the three conditions: (1) reflexive: every linear object $l$ is parallel to itself; (2) symmetric: if the linear object $l_1$ is parallel to the linear object $l_2$, then $l_2$ is $l_1$; (3) transitive: if the linear object $l_1$ is parallel to the linear object $l_2$, and if the linear object $l_2$ is parallel to the linear object $l_3$, then $l_1$ is parallel to $l_3$. -/
-instance (α : Type*) [ProjObj α] : IsEquiv α parallel where
+instance (α : Type*) [ProjObj α] : IsEquiv α Parallel where
   refl _ := rfl
   symm _ _ := Eq.symm
   trans _ _ _ := Eq.trans
 
-/-- This is to rewrite \verb|parallel l l'| as \verb|l ParallelTo l'| -/
-scoped infix : 50 " ParallelTo " => parallel
-
-/-- This is to rewrite \verb|parallel l l'| as $l \parallel l'$. -/
-scoped infix : 50 " ∥ " => parallel
-
 /- lots of trivial parallel relation of vec of 2 pt lies on Line, coercions, ... -/
 
 section coercion_theorem
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular.lean
index 391f7ae9..2c707e23 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular.lean
@@ -8,12 +8,12 @@ open Line
 variable {P : Type _} [EuclideanPlane P] {α β γ : Type*} [ProjObj α] [ProjObj β] [ProjObj γ]
   {l₁ : α} {l₂ : β} {l₃ : γ}
 
-def perpendicular (l₁ : α) (l₂ : β) : Prop :=
+def Perpendicular (l₁ : α) (l₂ : β) : Prop :=
   ProjObj.toProj l₁ = (ProjObj.toProj l₂).perp
 
-scoped infix : 50 " IsPerpendicularTo " => perpendicular
+scoped infix : 50 " IsPerpendicularTo " => Perpendicular
 
-scoped infix : 50 " ⟂ " => perpendicular
+scoped infix : 50 " ⟂ " => Perpendicular
 
 section Notation
 open Lean
@@ -21,34 +21,40 @@ open Lean
 syntax (name := perpendicularNotation) (priority := high) term:50 " ⟂ " term:51 : term
 
 @[macro perpendicularNotation] def perpendicularNotationImpl : Macro
-  | `($l:term ⟂ $r:term) => `(perpendicular $l $r)
+  | `($l:term ⟂ $r:term) => `(Perpendicular $l $r)
   | _ => Macro.throwUnsupported
 
 end Notation
 
-namespace perpendicular
+namespace Perpendicular
 
 @[simp]
 protected theorem irrefl (l : α) : ¬ (l ⟂ l) :=
   sorry
 
 protected theorem symm (h : l₁ ⟂ l₂) : (l₂ ⟂ l₁) := by
-  rw [perpendicular, Proj.perp, h, Proj.perp, vadd_vadd]
+  rw [Perpendicular, Proj.perp, h, Proj.perp, vadd_vadd]
   norm_cast
   simp
 
-end perpendicular
+end Perpendicular
 
 section Perpendicular_and_parallel
 
 theorem parallel_of_perp_perp (h₁ : l₁ ⟂ l₂) (h₂ : l₂ ⟂ l₃) : l₁ ∥ l₃ := by
-  unfold perpendicular at h₂
-  simp only [perpendicular, h₂, Proj.perp_perp] at h₁
+  unfold Perpendicular at h₂
+  simp only [Perpendicular, h₂, Proj.perp_perp] at h₁
   exact h₁
 
-theorem perp_of_parallel_perp (h₁ : l₁ ∥ l₂) (h₂ : l₂ ⟂ l₃) : l₁ ⟂ l₃ := h₁.trans h₂
+alias Perpendicular.trans := parallel_of_perp_perp
 
-theorem perp_of_perp_parallel (h₁ : l₁ ⟂ l₂) (h₂ : l₂ ∥ l₃) : l₁ ⟂ l₃ := h₁.trans (congrArg Proj.perp h₂)
+theorem perp_of_parallel_perp (h₁ : l₁ ∥ l₂) (h₂ : l₂ ⟂ l₃) : l₁ ⟂ l₃ := Eq.trans h₁ h₂
+
+alias Parallel.trans_perp := perp_of_parallel_perp
+
+theorem perp_of_perp_parallel (h₁ : l₁ ⟂ l₂) (h₂ : l₂ ∥ l₃) : l₁ ⟂ l₃ := Eq.trans h₁ (congrArg Proj.perp h₂)
+
+alias Perpendicular.trans_parallel := perp_of_perp_parallel
 
 theorem toProj_ne_toProj_of_perp (h : l₁ ⟂ l₂) : ProjObj.toProj l₁ ≠ ProjObj.toProj l₂ :=
   sorry
@@ -68,7 +74,7 @@ theorem toProj_of_perp_line_eq_toProj_perp (A : P) (l : Line P) : (perp_line A l
 theorem perp_line_perp (A : P) (l : Line P) : perp_line A l ⟂ l := toProj_of_perp_line_eq_toProj_perp A l
 
 theorem toProj_ne_perp_toProj (A : P) (l : Line P) : l.toProj ≠ (perp_line A l).toProj :=
-  Ne.trans_eq (perpendicular.irrefl l) (toProj_of_perp_line_eq_toProj_perp A l).symm
+  Ne.trans_eq (Perpendicular.irrefl l) (toProj_of_perp_line_eq_toProj_perp A l).symm
 
 def perp_foot (A : P) (l : Line P) : P := Line.inx l (perp_line A l) (toProj_ne_perp_toProj A l)
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular_trash.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular_trash.lean
index 8550b9a6..165c8ef0 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular_trash.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Perpendicular_trash.lean
@@ -12,7 +12,7 @@ theorem segnd_perp_line_of_line_perp_line {A B : P} (B_ne_A : B ≠ A) {l : Line
 theorem perp_foot_unique' {A B : P} {l : DirLine P} (h : A LiesOn l) (hne : B ≠ A) (hp : (DLIN B A hne.symm) ⟂ l) : perp_foot B l = A := sorry
 
 
-theorem not_collinear_with_perp_foot_of_ne_perp_foot (A B C : P) (l : Line P) (B_on_l : B LiesOn l) (A_not_on_l : ¬ A LiesOn l) (C_is_perp_foot : C = (perp_foot A l)) (B_ne_C : B ≠ C): ¬ collinear C B A := sorry
+theorem not_collinear_with_perp_foot_of_ne_perp_foot (A B C : P) (l : Line P) (B_on_l : B LiesOn l) (A_not_on_l : ¬ A LiesOn l) (C_is_perp_foot : C = (perp_foot A l)) (B_ne_C : B ≠ C): ¬ Collinear C B A := sorry
 
 theorem perp_foot_unique {A B : P} {l : DirLine P} (h : A LiesOn l) [_hne : PtNe A B] (hp : (DLIN B A) ⟂ l) : perp_foot B l = A := sorry
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Ratio.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Ratio.lean
index 43d6bf56..4b8886be 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Ratio.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Ratio.lean
@@ -12,16 +12,16 @@ section ratio
 /-
 Below is the definition of divratio using ddist, which I think might not be a good idea.
 
-def divratio (A B C : P) (colin : collinear A B C) (cnea : C ≠ A) : ℝ := (DirLine.ddist (B := B) (DirLine.pt_pt_maximal cnea (flip_collinear_snd_trd (triv_collinear A C))) (DirLine.pt_pt_maximal cnea (flip_collinear_snd_trd colin))) / (DirLine.ddist (DirLine.pt_pt_maximal cnea (flip_collinear_snd_trd (triv_collinear A C))) (DirLine.pt_pt_maximal cnea (perm_collinear_trd_fst_snd (triv_collinear C A))))
+def divratio (A B C : P) (colin : Collinear A B C) (cnea : C ≠ A) : ℝ := (DirLine.ddist (B := B) (DirLine.pt_pt_maximal cnea (Collinear.perm₁₃₂ (triv_collinear₁₂ A C))) (DirLine.pt_pt_maximal cnea (Collinear.perm₁₃₂ colin))) / (DirLine.ddist (DirLine.pt_pt_maximal cnea (Collinear.perm₁₃₂ (triv_collinear₁₂ A C))) (DirLine.pt_pt_maximal cnea (Collinear.perm₃₁₂ (triv_collinear₁₂ C A))))
 -/
 
 def divratio (A B C : P) : ℝ := ((VEC A B)/(VEC A C)).1
 
-theorem ratio_is_real' (A B C : P) (colin : collinear A B C) [cnea : PtNe C A] : ((VEC A B)/(VEC A C)).2 = 0 := by
-  have h0 : collinear A C B := flip_collinear_snd_trd colin
+theorem ratio_is_real' (A B C : P) (colin : Collinear A B C) [cnea : PtNe C A] : ((VEC A B)/(VEC A C)).2 = 0 := by
+  have h0 : Collinear A C B := Collinear.perm₁₃₂ colin
   rw [collinear_iff_eq_smul_vec_of_ne] at h0
   rcases h0 with ⟨r , h1⟩
-  have h2 : VEC A C ≠ 0 := (ne_iff_vec_ne_zero A C).mp cnea.out
+  have h2 : VEC A C ≠ 0 := ne_iff_vec_ne_zero.mp cnea.out
   rw [h1]
   calc
     (r • VEC A C / VEC A C).im = ((r : ℂ) • VEC A C / VEC A C).im := rfl
@@ -29,21 +29,21 @@ theorem ratio_is_real' (A B C : P) (colin : collinear A B C) [cnea : PtNe C A] :
       rw [Vec.smul_cdiv_cancel _ h2]
       rfl
 
-theorem ratio_is_real (A B C : P) (colin : collinear A B C) [cnea : PtNe C A] : (VEC A B)/(VEC A C) = divratio A B C := by
+theorem ratio_is_real (A B C : P) (colin : Collinear A B C) [cnea : PtNe C A] : (VEC A B)/(VEC A C) = divratio A B C := by
   have h0 : (divratio A B C : ℂ).re = ((VEC A B)/(VEC A C)).re := rfl
   have h1 : (divratio A B C : ℂ).im = ((VEC A B)/(VEC A C)).im := by
     rw [ratio_is_real' A B C colin]
     simp
   exact Complex.ext h0 h1.symm
 
-theorem vec_eq_vec_smul_ratio (A B C : P) (colin : collinear A B C) [cnea : PtNe C A] : VEC A B = (divratio A B C) • (VEC A C) := by
-  have h0 : VEC A C ≠ 0 := (ne_iff_vec_ne_zero A C).mp cnea.out
+theorem vec_eq_vec_smul_ratio (A B C : P) (colin : Collinear A B C) [cnea : PtNe C A] : VEC A B = (divratio A B C) • (VEC A C) := by
+  have h0 : VEC A C ≠ 0 := ne_iff_vec_ne_zero.mp cnea.out
   have h1 : VEC A B = ((VEC A B) / (VEC A C)) • VEC A C := by
     field_simp
   rw [h1, ratio_is_real A B C colin]
   field_simp
 
-theorem ratio_eq_ratio_div_ratio_minus_one (A B C : P) [cnea : PtNe C A] (colin : collinear A B C) : divratio B A C = (divratio A B C) / (divratio A B C - 1) := by
+theorem ratio_eq_ratio_div_ratio_minus_one (A B C : P) [cnea : PtNe C A] (colin : Collinear A B C) : divratio B A C = (divratio A B C) / (divratio A B C - 1) := by
   have h0 : VEC B A = (-divratio A B C) • VEC A C := by
     rw [← neg_vec A B, vec_eq_vec_smul_ratio A B C colin]
     field_simp
@@ -58,7 +58,7 @@ theorem ratio_eq_ratio_div_ratio_minus_one (A B C : P) [cnea : PtNe C A] (colin
     rw[h3]
   have h4 : VEC B A / VEC B C = (((divratio A B C / (divratio A B C - 1)) : ℝ ) : ℂ) := by
     rw [h0, h1]
-    have h5 : VEC A C ≠ 0 := (ne_iff_vec_ne_zero A C).mp cnea.out
+    have h5 : VEC A C ≠ 0 := ne_iff_vec_ne_zero.mp cnea.out
     field_simp
     norm_cast
     rw [Vec.neg_cdiv, Vec.smul_cdiv, Vec.cdiv_self h5, neg_div, ← div_neg]
@@ -69,11 +69,11 @@ theorem ratio_eq_ratio_div_ratio_minus_one (A B C : P) [cnea : PtNe C A] (colin
   rw [h4]
   rw [Complex.ofReal_re]
 
-theorem pt_eq_of_ratio_eq_of_ne_ne (A B C D : P) [cned : PtNe C D] [dnea : PtNe D A] [dneb : PtNe D B] (colinacd : collinear A C D) (colinbcd : collinear B C D) (req : divratio A C D = divratio B C D) : A = B := by
+theorem pt_eq_of_ratio_eq_of_ne_ne (A B C D : P) [cned : PtNe C D] [dnea : PtNe D A] [dneb : PtNe D B] (colinacd : Collinear A C D) (colinbcd : Collinear B C D) (req : divratio A C D = divratio B C D) : A = B := by
   have h0 : divratio C A D = divratio C B D := by
     rw [ratio_eq_ratio_div_ratio_minus_one A C D colinacd, req, ratio_eq_ratio_div_ratio_minus_one B C D colinbcd]
   have h1 : VEC C A = VEC C B := by
-    rw [vec_eq_vec_smul_ratio C A D (flip_collinear_fst_snd colinacd), vec_eq_vec_smul_ratio C B D (flip_collinear_fst_snd colinbcd), h0]
+    rw [vec_eq_vec_smul_ratio C A D (Collinear.perm₂₁₃ colinacd), vec_eq_vec_smul_ratio C B D (Collinear.perm₂₁₃ colinbcd), h0]
   rw [← start_vadd_vec_eq_end C A, h1, start_vadd_vec_eq_end C B]
 
 theorem ratio_eq_zero_of_point_eq1 (A B : P) : divratio A A B = 0 := by
diff --git a/EuclideanGeometry/Foundation/Axiom/Linear/Ray.lean b/EuclideanGeometry/Foundation/Axiom/Linear/Ray.lean
index f2aa651b..32ecf7e4 100644
--- a/EuclideanGeometry/Foundation/Axiom/Linear/Ray.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Linear/Ray.lean
@@ -84,7 +84,6 @@ end Seg
 /-- A \emph{nondegenerate segment} is a segment $AB$ that is nondegenerate, i.e. $A \neq B$. -/
 def SegND (P : Type _) [EuclideanPlane P] := {seg : Seg P // seg.IsND}
 
-
 end definition
 
 variable {P : Type _} [EuclideanPlane P]
@@ -241,9 +240,11 @@ abbrev source (seg_nd : SegND P) : P := seg_nd.1.source
 @[pp_dot]
 abbrev target (seg_nd : SegND P) : P := seg_nd.1.target
 
+instance {s : SegND P} : PtNe s.target s.source := ⟨s.2⟩
+
 /-- Given a nondegenerate segment $AB$, this function returns the nondegenerate vector $\overrightarrow{AB}$. -/
 @[pp_dot]
-def toVecND (seg_nd : SegND P) : VecND := ⟨VEC seg_nd.source seg_nd.target, (ne_iff_vec_ne_zero _ _).mp seg_nd.2⟩
+def toVecND (seg_nd : SegND P) : VecND := ⟨VEC seg_nd.source seg_nd.target, ne_iff_vec_ne_zero.mp seg_nd.2⟩
 
 /-- Given a nondegenerate segment $AB$, this function returns the direction associated to the segment, defined by normalizing the nondegenerate vector $\overrightarrow{AB}$. -/
 @[pp_dot]
@@ -804,8 +805,15 @@ theorem nd_of_rev_of_nd {seg : Seg P} (nd : seg.IsND) : seg.reverse.IsND := by
   apply nd
 
 /-- Given a nondegenerate segment $AB$, this function returns the reversed nondegenerate segment $BA$. -/
+@[pp_dot]
 def SegND.reverse (seg_nd : SegND P) : SegND P := ⟨seg_nd.1.reverse, nd_of_rev_of_nd seg_nd.2⟩
 
+@[simp]
+lemma SegND.reverse_source (seg_nd : SegND P) : seg_nd.reverse.source = seg_nd.target := rfl
+
+@[simp]
+lemma SegND.reverse_target (seg_nd : SegND P) : seg_nd.reverse.target = seg_nd.source := rfl
+
 /-- The reverse of a nondegenerate segment $AB$ is the nondegenerate segment $BA$. -/
 @[simp]
 theorem seg_nd_rev {A B : P} [_h : PtNe B A] : (SEG_nd A B).reverse = SEG_nd B A := rfl
@@ -1182,7 +1190,7 @@ theorem target_lies_int_seg_source_pt_of_pt_lies_int_extn {X : P} {seg_nd : SegN
     have: A=0:=by linarith
     rw [this] at ha
     simp only [Dir.toVec_neg_eq_neg_toVec, smul_neg, zero_smul, neg_zero] at ha
-    apply (eq_iff_vec_eq_zero _ _).mpr
+    apply eq_iff_vec_eq_zero.mpr
     exact ha
   have raysourcesource:seg_nd.extension.source=seg_nd.1.target:=by
     rfl
@@ -1538,7 +1546,7 @@ theorem SegND.exist_pt_beyond_pt (l : SegND P) : (∃ q : P, l.target LiesInt (S
     If a segment contains an interior point, then it is nondegenerate-/
 theorem Seg.nd_of_exist_int_pt {X : P} {seg : Seg P} (h : X LiesInt seg) : seg.IsND := by
   rcases h with ⟨⟨_, ⟨_, _, e⟩⟩, p_ne_s, _⟩
-  have t : VEC seg.source X ≠ 0 := (ne_iff_vec_ne_zero seg.source X).mp p_ne_s
+  have t : VEC seg.source X ≠ 0 := ne_iff_vec_ne_zero.mp p_ne_s
   rw [e] at t
   exact Iff.mp vsub_ne_zero (right_ne_zero_of_smul t)
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Position/Angle.lean b/EuclideanGeometry/Foundation/Axiom/Position/Angle.lean
index 4f04c63f..e094d8d0 100644
--- a/EuclideanGeometry/Foundation/Axiom/Position/Angle.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Position/Angle.lean
@@ -618,19 +618,19 @@ theorem dir_perp_iff_isRight : ang.dir₁ ⟂ ang.dir₂ ↔ ang.IsRight :=
 theorem value_eq_pi_of_lies_int_seg_nd {A B C : P} [PtNe C A] (h : B LiesInt (SEG_nd A C)) : ∠ A B C h.2.symm h.3.symm = π :=
   value_eq_pi_of_eq_neg_dir ((SEG_nd A C).toDir_eq_neg_toDir_of_lies_int h)
 
-theorem collinear_iff_dvalue_eq_zero : collinear O A B ↔ (ANG A O B).dvalue = 0 :=
-  collinear_iff_toProj_eq_of_ptNe.trans (eq_comm.trans vsub_eq_zero_iff_eq.symm)
+theorem collinear_iff_dvalue_eq_zero : Collinear O A B ↔ (ANG A O B).dvalue = 0 :=
+  collinear_iff_toProj_eq_of_ptNe.trans vsub_eq_zero_iff_eq.symm
 
-theorem collinear_iff_not_isND : collinear O A B ↔ ¬ (ANG A O B).IsND :=
+theorem collinear_iff_not_isND : Collinear O A B ↔ ¬ (ANG A O B).IsND :=
   collinear_iff_dvalue_eq_zero.trans not_isND_iff_coe.symm
 
-theorem not_collinear_iff_isND : ¬ collinear O A B ↔ (ANG A O B).IsND :=
+theorem not_collinear_iff_isND : ¬ Collinear O A B ↔ (ANG A O B).IsND :=
   collinear_iff_not_isND.not.trans not_not
 
-theorem collinear_of_angle_eq_zero (h : ∠ A O B = 0) : collinear O A B :=
+theorem collinear_of_angle_eq_zero (h : ∠ A O B = 0) : Collinear O A B :=
   collinear_iff_not_isND.mpr (not_isND_of_eq_zero h)
 
-theorem collinear_of_angle_eq_pi (h : ∠ A O B = π ) : collinear O A B :=
+theorem collinear_of_angle_eq_pi (h : ∠ A O B = π ) : Collinear O A B :=
   collinear_iff_not_isND.mpr (not_isND_of_eq_pi h)
 
 end angle_value
diff --git a/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex.lean b/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex.lean
index aec5dfbf..500dc928 100644
--- a/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex.lean
@@ -51,7 +51,7 @@ theorem IsND_of_oppo_of_IsND (nontriv : ang.IsND) : ang.oppo.IsND := by sorry
 
 end Angle
 
-section parallel
+section Parallel
 variable {P : Type _} [EuclideanPlane P]
 -- should be stated use mod 2pi first, then back to pi or -pi
 
@@ -90,7 +90,7 @@ theorem value_eq_of_isalternateintang {ang₁ ang₂ : Angle P} (h : IsAlternate
 theorem value_eq_of_corresponding_angle {l₁ l₂ l : DirLine P} (h : l₁.toDir = l₂.toDir) (g : ¬ l ∥ l₁) : (Angle.mk_dirline_dirline l₁ l (Ne.symm g)).value = (Angle.mk_dirline_dirline l₂ l (Ne.symm (ne_of_ne_of_eq g (Quotient.sound (h ▸ PM.con.refl _))))).value := sorry
 -/
 
-end parallel
+end Parallel
 
 namespace Angle
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex_trash.lean b/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex_trash.lean
index d7967ef4..b22e6eb7 100644
--- a/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex_trash.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Position/Angle_ex_trash.lean
@@ -8,9 +8,9 @@ variable {P : Type _} [EuclideanPlane P]
 
 theorem perp_foot_lies_int_ray_of_acute_ang {A B C : P} (b_ne_a : B ≠ A) (c_ne_a : C ≠ A) (acu : Angle.IsAcu (ANG B A C b_ne_a c_ne_a)) : (perp_foot C (LIN A B b_ne_a)) LiesInt (RAY A B b_ne_a) := by sorry
 
-theorem ang_acute_of_is_isoceles {A B C : P} (not_collinear_ABC : ¬ collinear A B C) (isoceles_ABC : (▵ A B C).IsIsoceles) : Angle.IsAcu (ANG C B A (ne_of_not_collinear not_collinear_ABC).1 (ne_of_not_collinear not_collinear_ABC).2.2.symm) := by sorry
+theorem ang_acute_of_is_isoceles {A B C : P} (not_collinear_ABC : ¬ Collinear A B C) (isoceles_ABC : (▵ A B C).IsIsoceles) : Angle.IsAcu (ANG C B A (ne_of_not_collinear not_collinear_ABC).1 (ne_of_not_collinear not_collinear_ABC).2.2.symm) := by sorry
 
-theorem ang_acute_of_is_isoceles_variant {A B C : P} (not_collinear_ABC : ¬ collinear A B C) (isoceles_ABC : (▵ A B C).IsIsoceles) : Angle.IsAcu (ANG A C B (ne_of_not_collinear not_collinear_ABC).2.1 (ne_of_not_collinear not_collinear_ABC).1.symm) := by sorry
+theorem ang_acute_of_is_isoceles_variant {A B C : P} (not_collinear_ABC : ¬ Collinear A B C) (isoceles_ABC : (▵ A B C).IsIsoceles) : Angle.IsAcu (ANG A C B (ne_of_not_collinear not_collinear_ABC).2.1 (ne_of_not_collinear not_collinear_ABC).1.symm) := by sorry
 
 theorem is_acute_of_is_acute_rev {O A B : P} (h1 : A ≠ O) (h2 : B ≠ O) (h3 : Angle.IsAcu (ANG A O B h1 h2)) : Angle.IsAcu (ANG B O A h2 h1) := by sorry
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Position/Orientation.lean b/EuclideanGeometry/Foundation/Axiom/Position/Orientation.lean
index 02561e16..3a47643e 100644
--- a/EuclideanGeometry/Foundation/Axiom/Position/Orientation.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Position/Orientation.lean
@@ -1,7 +1,4 @@
-import EuclideanGeometry.Foundation.Axiom.Linear.Collinear
-import EuclideanGeometry.Foundation.Axiom.Linear.Parallel
 import EuclideanGeometry.Foundation.Axiom.Position.Angle
-import EuclideanGeometry.Foundation.Axiom.Position.Angle_trash
 import EuclideanGeometry.Foundation.Axiom.Linear.Line_trash
 
 /- This file discuss the relative positions of points and rays on a plane. -/
@@ -12,30 +9,8 @@ open Classical AngValue Angle
 
 variable {P : Type*} [EuclideanPlane P]
 
-/- Definition of the wedge of three points.-/
-
 section wedge
 
-def wedge (A B C : P) : ℝ := Vec.det (VEC A B) (VEC A C)
-
-def oarea (A B C : P) : ℝ := wedge A B C / 2
-
-theorem wedge213 (A B C : P) : wedge B A C = - wedge A B C := by
-  unfold wedge
-  rw [← neg_vec A B,← vec_sub_vec A B C, map_sub]
-  simp only [map_neg, LinearMap.neg_apply, Vec.det_self, neg_zero, sub_zero]
-
-theorem wedge132 (A B C : P) : wedge A C B = - wedge A B C := by
-  unfold wedge
-  rw [Vec.det_skew]
-
-theorem wedge312 (A B C : P) : wedge C A B = wedge A B C := by
-  rw [wedge213, wedge132, neg_neg]
-
-theorem wedge231 (A B C : P) : wedge B C A = wedge A B C := by rw [wedge312, wedge312]
-
-theorem wedge321 (A B C : P) : wedge C B A = - wedge A B C := by rw [wedge213, wedge231]
-
 theorem wedge_eq_length_mul_length_mul_sin (A B C : P) [bnea : PtNe B A] [cnea : PtNe C A] : wedge A B C = (SEG A B).length * (SEG A C).length * sin (ANG B A C).value := by
   unfold wedge
   have h0 : (ANG B A C).value = VecND.angle (VEC_nd A B) (VEC_nd A C)  := rfl
@@ -43,32 +18,7 @@ theorem wedge_eq_length_mul_length_mul_sin (A B C : P) [bnea : PtNe B A] [cnea :
   rw [Seg.length_eq_norm_toVec, Seg.length_eq_norm_toVec]
   exact (VecND.norm_mul_sin (VEC_nd A B) (VEC_nd A C)).symm
 
-theorem collinear_iff_wedge_eq_zero (A B C : P) : (collinear A B C) ↔ (wedge A B C = 0) := by
-  dsimp only [wedge]
-  by_cases h : PtNe B A
-  · have vecabnd : VEC A B ≠ 0 := by
-      exact (ne_iff_vec_ne_zero A B).mp h.out
-    rw [← Vec.det_skew, neg_eq_zero, Vec.det_eq_zero_iff_eq_smul_right]
-    simp only [vecabnd, false_or]
-    constructor
-    · intro k
-      exact collinear_iff_eq_smul_vec_of_ne.mp k
-    · intro k
-      exact collinear_iff_eq_smul_vec_of_ne.mpr k
-  · simp [PtNe, fact_iff] at h
-    have vecab0 : VEC A B = 0 := by
-      exact (eq_iff_vec_eq_zero A B).mp h
-    constructor
-    intro
-    field_simp [vecab0]
-    intro
-    rw [h]
-    exact triv_collinear A C
-
-theorem not_collinear_iff_wedge_ne_zero (A B C : P) : (¬ collinear A B C) ↔ (wedge A B C ≠ 0) := by
-  rw [collinear_iff_wedge_eq_zero]
-
-theorem wedge_pos_iff_angle_pos (A B C : P) (nd : ¬collinear A B C) : (0 < wedge A B C) ↔ (Angle.mk_pt_pt_pt B A C (ne_of_not_collinear nd).2.2 (ne_of_not_collinear nd).2.1.symm).value.IsPos := by
+theorem wedge_pos_iff_angle_pos (A B C : P) (nd : ¬Collinear A B C) : (0 < wedge A B C) ↔ (Angle.mk_pt_pt_pt B A C (ne_of_not_collinear nd).2.2 (ne_of_not_collinear nd).2.1.symm).value.IsPos := by
   have h1 : 0 < dist A B := by
       have abnd : (SEG A B).IsND := (ne_of_not_collinear nd).2.2
       exact dist_pos.mpr (abnd.symm)
@@ -112,9 +62,9 @@ theorem odist'_eq_length_mul_sin (A : P) (ray : Ray P) [h : PtNe A ray.source] :
 
 theorem wedge_eq_length_mul_odist' (A B C : P) [bnea : PtNe B A] : (wedge A B C) = (SEG A B).length * odist' C (RAY A B) := by
   by_cases p : C = A
-  · have vecrayc0 : VEC (RAY A B).source C = 0 := (eq_iff_vec_eq_zero A C).mp p
+  · have vecrayc0 : VEC (RAY A B).source C = 0 := eq_iff_vec_eq_zero.mp p
     dsimp only [wedge, odist']
-    field_simp [(eq_iff_vec_eq_zero A C).mp p, vecrayc0]
+    field_simp [eq_iff_vec_eq_zero.mp p, vecrayc0]
   · haveI cnea : PtNe C A := ⟨p⟩
     rw [wedge_eq_length_mul_length_mul_sin,
       @odist'_eq_length_mul_sin _ _ C (RAY A B) cnea, ← mul_assoc]
@@ -164,7 +114,7 @@ theorem odist_reverse_eq_neg_odist {α} (A : P) [DirFig α P] (df : α) : odist
   rw [odist_reverse_eq_neg_odist'' A (toDirLine df)]
   rfl
 
-theorem wedge_eq_wedge_iff_odist_eq_odist_of_ne (A B C D : P) [bnea : PtNe B A] : (odist C (SEG_nd A B) = odist D (SEG_nd A B)) ↔ (wedge A B C = wedge A B D) := by
+theorem odist_eq_odist_iff_wedge_eq_wedge (A B C D : P) [bnea : PtNe B A] : (odist C (SEG_nd A B) = odist D (SEG_nd A B)) ↔ (wedge A B C = wedge A B D) := by
   rw [wedge_eq_length_mul_odist' A B C, wedge_eq_length_mul_odist' A B D]
   field_simp
   tauto
@@ -335,7 +285,7 @@ theorem LiesOnLeft_or_LiesOnRight_of_not_LiesOn {A : P} [DirFig α P] {df : α}
   · exact lr
 
 
-theorem not_collinear_of_LiesOnLeft_or_LiesOnRight (A B C : P) [bnea : PtNe B A] (hlr : (IsOnLeftSide C (RAY A B)) ∨ (IsOnRightSide C (RAY A B))) : ¬ collinear A B C := by
+theorem not_collinear_of_LiesOnLeft_or_LiesOnRight (A B C : P) [bnea : PtNe B A] (hlr : (IsOnLeftSide C (RAY A B)) ∨ (IsOnRightSide C (RAY A B))) : ¬ Collinear A B C := by
   apply (not_collinear_iff_wedge_ne_zero A B C).mpr
   have hw : (wedge A B C) = (SEG A B).length * odist' C (RAY A B) := by
     exact wedge_eq_length_mul_odist' A B C
@@ -377,21 +327,21 @@ theorem oarea_eq_length_mul_odist_div_two (A B C : P) [bnea : PtNe B A] : oarea
   rw [h2] at h1
   rw [h1]
 
-theorem oarea_eq_oarea_iff_odist_eq_odist_of_ne (A B C D : P) [bnea : PtNe B A] : odist C (SEG_nd A B) = odist D (SEG_nd A B) ↔ oarea A B C = oarea A B D := by
+theorem odist_eq_odist_iff_oarea_eq_oarea (A B C D : P) [bnea : PtNe B A] : odist C (SEG_nd A B) = odist D (SEG_nd A B) ↔ oarea A B C = oarea A B D := by
   unfold oarea
   field_simp
-  exact wedge_eq_wedge_iff_odist_eq_odist_of_ne A B C D
+  exact odist_eq_odist_iff_wedge_eq_wedge A B C D
 
 theorem oarea_eq_sin_mul_length_mul_length_div_two (A B C : P) [bnea : PtNe B A] [cnea : PtNe C A] : oarea A B C = (SEG A B).length * (SEG A C).length * sin (ANG B A C).value / 2 := by
   unfold oarea
   rw [wedge_eq_length_mul_length_mul_sin A B C]
 
-theorem oarea_eq_zero_iff_collinear (A B C : P) : oarea A B C = 0 ↔ collinear A B C := by
+theorem oarea_eq_zero_iff_collinear (A B C : P) : oarea A B C = 0 ↔ Collinear A B C := by
   unfold oarea
   field_simp
   exact (collinear_iff_wedge_eq_zero A B C).symm
 
-theorem oarea_tri_nd_ne_zero (A B C : P) (trind : ¬ collinear A B C) : oarea A B C ≠ 0 := by
+theorem oarea_tri_nd_ne_zero (A B C : P) (trind : ¬ Collinear A B C) : oarea A B C ≠ 0 := by
   rw[← oarea_eq_zero_iff_collinear A B C] at trind
   tauto
 
@@ -399,12 +349,12 @@ end oriented_area
 
 section cooperation_with_parallel
 
-theorem odist_eq_odist_of_parallel' (A B : P) (ray : Ray P) [bnea : PtNe B A] (para : parallel (SEG_nd A B) ray) : odist A ray =odist B ray := by
+theorem odist_eq_odist_of_parallel' (A B : P) (ray : Ray P) [bnea : PtNe B A] (para : Parallel (SEG_nd A B) ray) : odist A ray =odist B ray := by
   unfold odist
   have h1 : Vec.det ray.2.unitVec (VEC ray.1 B) = Vec.det ray.2.unitVec (VEC ray.1 A) + Vec.det ray.2.unitVec (VEC A B)
   · rw [← map_add, ←vec_add_vec ray.1 A B]
   have h2 : Vec.det ray.2.unitVec (VEC A B) = 0
-  · unfold parallel at para
+  · unfold Parallel at para
     have h3 : Dir.toProj ray.2 = (VEC_nd A B).toProj := para.symm
     have h4 : VecND.toProj ray.2.unitVecND = (VEC_nd A B).toProj := by
       rw [← h3]
@@ -417,11 +367,11 @@ theorem odist_eq_odist_of_parallel' (A B : P) (ray : Ray P) [bnea : PtNe B A] (p
   rw [add_zero] at h1
   exact h1.symm
 
---theorem odist_eq_odist_of_parallel {α} [DirFig α P] (A B : P) (df : α) [bnea : PtNe B A] (para : parallel (SEG_nd A B) df) : odist A df = odist B df := sorry
+--theorem odist_eq_odist_of_parallel {α} [DirFig α P] (A B : P) (df : α) [bnea : PtNe B A] (para : Parallel (SEG_nd A B) df) : odist A df = odist B df := sorry
 
-theorem odist_eq_odist_iff_parallel_ne {α} [DirFig α P] (A B : P) (df : α) [bnea : PtNe B A] : (parallel (SEG_nd A B) df) ↔ odist A df = odist B df := by
-  have coer1 : (parallel (SEG_nd A B) df) = (parallel (SEG_nd A B) (toDirLine df)) := by
-    unfold parallel
+theorem parallel_iff_odist_eq_odist_right {α} [DirFig α P] (A B : P) (df : α) [bnea : PtNe B A] : Parallel (SEG_nd A B) df ↔ odist A df = odist B df := by
+  have coer1 : (Parallel (SEG_nd A B) df) = (Parallel (SEG_nd A B) (toDirLine df)) := by
+    unfold Parallel
     have : toProj df = toProj (toDirLine df) := by
       simp only [← DirObj.toDir_toProj_eq_toProj]
       congr
@@ -464,7 +414,7 @@ theorem odist_eq_odist_iff_parallel_ne {α} [DirFig α P] (A B : P) (df : α) [b
     simp only [eq_iff_iff]
     exact VecND.det_eq_zero_iff_toProj_eq_toProj (u := ray.toDir.unitVecND) (v := VEC_nd A B)
   simp only [h,h']
-  unfold parallel
+  unfold Parallel
   have : toProj (SEG_nd A B) = (VEC_nd A B).toProj := by rfl
   constructor
   · intro p
@@ -477,38 +427,33 @@ theorem odist_eq_odist_iff_parallel_ne {α} [DirFig α P] (A B : P) (df : α) [b
     simp only [Dir.unitVecND_toProj]
     rfl
 
-theorem wedge_eq_wedge_iff_parallel_of_ne_ne (A B C D : P) [bnea : PtNe B A] [dnec : PtNe D C] : (parallel (SEG_nd A B) (SEG_nd C D)) ↔ wedge A B C = wedge A B D := by
-  have : (wedge A B C = wedge A B D) = (odist C (SEG_nd A B) = odist D (SEG_nd A B)) := by
-    symm
-    simp only [eq_iff_iff]
-    exact wedge_eq_wedge_iff_odist_eq_odist_of_ne A B C D
-  simp only [this]
-  have : (parallel (SEG_nd A B) (SEG_nd C D)) = (parallel (SEG_nd C D) (SEG_nd A B)) := by
-    unfold parallel
-    simp only [eq_iff_iff]
-    constructor
-    · intro P
-      exact P.symm
-    · intro P
-      exact P.symm
-  simp only [this]
-  exact odist_eq_odist_iff_parallel_ne C D (SEG_nd A B)
-
-theorem oarea_eq_oarea_iff_parallel_ne (A B C D : P) [bnea : PtNe B A] [dnec : PtNe D C] : (parallel (SEG_nd A B) (SEG_nd C D)) ↔ oarea A B C = oarea A B D := by
+theorem wedge_eq_wedge_iff_parallel_left {A B C D : P} [bnea : PtNe B A] [dnec : PtNe D C] : wedge A B C = wedge A B D ↔ SEG_nd A B ∥ SEG_nd C D := by
+  rw [← odist_eq_odist_iff_wedge_eq_wedge, parallel_comm, parallel_iff_odist_eq_odist_right]
+
+theorem wedge_eq_wedge_iff_parallel_right {A B C D : P} [bnea : PtNe B A] [dnec : PtNe D C] : wedge C D A = wedge C D B ↔ SEG_nd A B ∥ SEG_nd C D := by
+  rw [← odist_eq_odist_iff_wedge_eq_wedge, parallel_iff_odist_eq_odist_right]
+
+theorem parallel_iff_wedge_eq_wedge_left {s₁ s₂ : SegND P} : s₁ ∥ s₂ ↔ wedge s₁.source s₁.target s₂.source = wedge s₁.source s₁.target s₂.target :=
+  wedge_eq_wedge_iff_parallel_left.symm
+
+theorem parallel_iff_wedge_eq_wedge_right {s₁ s₂ : SegND P} : s₁ ∥ s₂ ↔ wedge s₂.source s₂.target s₁.source = wedge s₂.source s₂.target s₁.target :=
+  wedge_eq_wedge_iff_parallel_right.symm
+
+theorem oarea_eq_oarea_iff_parallel_left {A B C D : P} [bnea : PtNe B A] [dnec : PtNe D C] : oarea A B C = oarea A B D ↔ SEG_nd A B ∥ SEG_nd C D := by
+  rw [← wedge_eq_wedge_iff_parallel_left]
   unfold oarea
-  have : (wedge A B C / 2 = wedge A B D / 2) = (wedge A B C = wedge A B D) := by
-    simp only [eq_iff_iff]
-    constructor
-    · intro P
-      calc
-        _= wedge A B C /2 *2 := by simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
-          div_mul_cancel]
-        _= wedge A B D /2 *2 := by simp only [P]
-        _=_ := by simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, div_mul_cancel]
-    · intro P
-      simp only [P]
-  simp only [this]
-  exact wedge_eq_wedge_iff_parallel_of_ne_ne A B C D
+  field_simp
+
+theorem oarea_eq_oarea_iff_parallel_right {A B C D : P} [bnea : PtNe B A] [dnec : PtNe D C] : oarea C D A = oarea C D B ↔ SEG_nd A B ∥ SEG_nd C D := by
+  rw [← wedge_eq_wedge_iff_parallel_right]
+  unfold oarea
+  field_simp
+
+theorem parallel_iff_oarea_eq_oarea_left {s₁ s₂ : SegND P} : s₁ ∥ s₂ ↔ oarea s₁.source s₁.target s₂.source = oarea s₁.source s₁.target s₂.target :=
+  oarea_eq_oarea_iff_parallel_left.symm
+
+theorem parallel_iff_oarea_eq_oarea_right {s₁ s₂ : SegND P} : s₁ ∥ s₂ ↔ oarea s₂.source s₂.target s₁.source = oarea s₂.source s₂.target s₁.target :=
+  oarea_eq_oarea_iff_parallel_right.symm
 
 end cooperation_with_parallel
 
@@ -517,7 +462,7 @@ scoped infix : 55 " LiesOnRight " => IsOnRightSide
 
 section handside
 
-theorem same_sign_of_parallel (A B : P) (ray : Ray P) [bnea : PtNe B A] (para : parallel (RAY A B)  ray) : odist_sign A ray = odist_sign B ray := by
+theorem same_sign_of_parallel (A B : P) (ray : Ray P) [bnea : PtNe B A] (para : Parallel (RAY A B)  ray) : odist_sign A ray = odist_sign B ray := by
   unfold odist_sign
   rw [odist_eq_odist_of_parallel' A B ray para]
 
@@ -1067,7 +1012,7 @@ theorem LiesOnLeft_iff_LiesOnLeft_rev_of_IsOnOppositeSide (A B : P) (dl : DirLin
     simp [this] at h0
     exact h0
 
-theorem not_collinear_of_IsOnSameSide (A B C D : P) [bnea : PtNe B A] (h : IsOnSameSide C D (RAY A B)) : (¬ collinear A B C) ∧ (¬ collinear A B D) := by
+theorem not_collinear_of_IsOnSameSide (A B C D : P) [bnea : PtNe B A] (h : IsOnSameSide C D (RAY A B)) : (¬ Collinear A B C) ∧ (¬ Collinear A B D) := by
   have hlr : (IsOnLeftSide C (RAY A B)) ∨ (IsOnRightSide C (RAY A B)) := by
     rcases h with l|r
     · simp only [l.1, true_or]
@@ -1076,15 +1021,15 @@ theorem not_collinear_of_IsOnSameSide (A B C D : P) [bnea : PtNe B A] (h : IsOnS
     rcases h with l|r
     · simp only [l.2, true_or]
     · simp only [r.2, or_true]
-  have c : ¬ collinear A B C := by
+  have c : ¬ Collinear A B C := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     exact hlr
-  have d : ¬ collinear A B D := by
+  have d : ¬ Collinear A B D := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     exact hlr'
   simp only [c, not_false_eq_true, d, and_self]
 
-theorem not_collinear_of_IsOnOppositeSide (A B C D : P) [bnea : PtNe B A] (h : IsOnOppositeSide C D (RAY A B)) : (¬ collinear A B C) ∧ (¬ collinear A B D) := by
+theorem not_collinear_of_IsOnOppositeSide (A B C D : P) [bnea : PtNe B A] (h : IsOnOppositeSide C D (RAY A B)) : (¬ Collinear A B C) ∧ (¬ Collinear A B D) := by
   have hlr : (IsOnLeftSide C (RAY A B)) ∨ (IsOnRightSide C (RAY A B)) := by
     rcases h with l|r
     · simp only [l.1, true_or]
@@ -1093,10 +1038,10 @@ theorem not_collinear_of_IsOnOppositeSide (A B C D : P) [bnea : PtNe B A] (h : I
     rcases h with l|r
     · simp only [l.2, or_true]
     · simp only [r.2, true_or]
-  have c : ¬ collinear A B C := by
+  have c : ¬ Collinear A B C := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     exact hlr
-  have d : ¬ collinear A B D := by
+  have d : ¬ Collinear A B D := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     exact hlr'
   simp only [c, not_false_eq_true, d, and_self]
diff --git a/EuclideanGeometry/Foundation/Axiom/Triangle/Basic.lean b/EuclideanGeometry/Foundation/Axiom/Triangle/Basic.lean
index a2b52b95..b2a01792 100644
--- a/EuclideanGeometry/Foundation/Axiom/Triangle/Basic.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Triangle/Basic.lean
@@ -38,7 +38,7 @@ def oarea (tr : Triangle P) : ℝ := EuclidGeom.oarea tr.1 tr.2 tr.3
 def area (tr : Triangle P) : ℝ := |tr.oarea|
 
 @[pp_dot]
-def IsND (tr : Triangle P) : Prop := ¬ collinear tr.1 tr.2 tr.3
+def IsND (tr : Triangle P) : Prop := ¬ Collinear tr.1 tr.2 tr.3
 
 end Triangle
 
@@ -73,13 +73,13 @@ abbrev edge₂ : Seg P := tr_nd.1.edge₂
 abbrev edge₃ : Seg P := tr_nd.1.edge₃
 
 @[pp_dot]
-def edge_nd₁ : SegND P := ⟨tr_nd.1.edge₁, tr_nd.nontriv₁.out⟩
+def edgeND₁ : SegND P := ⟨tr_nd.1.edge₁, tr_nd.nontriv₁.out⟩
 
 @[pp_dot]
-def edge_nd₂ : SegND P := ⟨tr_nd.1.edge₂, tr_nd.nontriv₂.out⟩
+def edgeND₂ : SegND P := ⟨tr_nd.1.edge₂, tr_nd.nontriv₂.out⟩
 
 @[pp_dot]
-def edge_nd₃ : SegND P := ⟨tr_nd.1.edge₃, tr_nd.nontriv₃.out⟩
+def edgeND₃ : SegND P := ⟨tr_nd.1.edge₃, tr_nd.nontriv₃.out⟩
 
 @[pp_dot]
 abbrev oarea : ℝ := tr_nd.1.oarea
@@ -108,7 +108,7 @@ namespace Triangle
 
 -- When we have DirFig, rewrite this definition.
 protected def IsInt (A : P) (tr : Triangle P) : Prop := by
-  by_cases h : collinear tr.1 tr.2 tr.3
+  by_cases h : Collinear tr.1 tr.2 tr.3
   -- why not using ¬ tr.IsND?
   · exact False
   · let tr_nd : TriangleND P := ⟨tr, h⟩
@@ -192,9 +192,9 @@ end Triangle
 
 namespace TriangleND
 
-def perm_vertices : (TriangleND P) := ⟨tr_nd.1.perm_vertices, flip_collinear_snd_trd.mt $ flip_collinear_fst_snd.mt tr_nd.2⟩
+def perm_vertices : (TriangleND P) := ⟨tr_nd.1.perm_vertices, Collinear.perm₁₃₂.mt $ Collinear.perm₂₁₃.mt tr_nd.2⟩
 
-def flip_vertices : (TriangleND P) := ⟨tr_nd.1.flip_vertices, flip_collinear_snd_trd.mt tr_nd.2⟩
+def flip_vertices : (TriangleND P) := ⟨tr_nd.1.flip_vertices, Collinear.perm₁₃₂.mt tr_nd.2⟩
 
 theorem eq_self_of_perm_vertices_three_times : tr_nd.perm_vertices.perm_vertices.perm_vertices = tr_nd := rfl
   --exact tr_nd.1.eq_self_of_perm_vertices_three_times
@@ -212,7 +212,7 @@ theorem same_orient_of_perm_vertices : tr_nd.is_cclock = (tr_nd.perm_vertices.is
   have eq : wedge tr_nd.perm_vertices.point₁ tr_nd.perm_vertices.point₂ tr_nd.perm_vertices.point₃ = wedge tr_nd.point₁ tr_nd.point₂ tr_nd.point₃ := by
     calc
       _= wedge tr_nd.point₂ tr_nd.point₃ tr_nd.point₁ := by rfl
-      _=_ := by exact wedge231 tr_nd.point₁ tr_nd.point₂ tr_nd.point₃
+      _=_ := by exact wedge312 tr_nd.point₁ tr_nd.point₂ tr_nd.point₃
   simp only [w1,w2,eq]
 
 theorem reverse_orient_of_flip_vertices : tr_nd.is_cclock = ¬ tr_nd.flip_vertices.is_cclock := by
@@ -230,7 +230,7 @@ theorem reverse_orient_of_flip_vertices : tr_nd.is_cclock = ¬ tr_nd.flip_vertic
   · intro P
     simp at P
     have ne0' : wedge tr_nd.point₁ tr_nd.point₂ tr_nd.point₃ ≠ 0 := by
-      have : ¬ collinear tr_nd.point₁ tr_nd.point₂ tr_nd.point₃ := by
+      have : ¬ Collinear tr_nd.point₁ tr_nd.point₂ tr_nd.point₃ := by
         exact tr_nd.2
       apply (collinear_iff_wedge_eq_zero tr_nd.point₁ tr_nd.point₂ tr_nd.point₃).not.mp
       exact this
@@ -245,7 +245,7 @@ theorem is_inside_of_is_inside_flip_vertices (tr_nd : Triangle P) (p : P) (insid
 
 end TriangleND
 
-def TriangleND.mk (A B C : P) (h : ¬ collinear A B C) : TriangleND P := Subtype.mk (Triangle.mk A B C) h
+def TriangleND.mk (A B C : P) (h : ¬ Collinear A B C) : TriangleND P := Subtype.mk (Triangle.mk A B C) h
 
 scoped notation "TRI" => Triangle.mk
 scoped notation "▵" => Triangle.mk
@@ -313,20 +313,20 @@ theorem trivial_of_edge_sum_eq_edge : tr.edge₁.length + tr.edge₂.length = tr
   unfold IsND
   rw [not_not]
   by_cases h₁ : VEC B C = 0
-  · simp only [(eq_iff_vec_eq_zero B C).2 h₁]
-    apply flip_collinear_fst_trd
-    exact triv_collinear _ _
+  · simp only [eq_iff_vec_eq_zero.2 h₁]
+    apply Collinear.perm₃₂₁
+    exact triv_collinear₁₂ _ _
   · by_cases h₂ : VEC C A = 0
-    · simp only [(eq_iff_vec_eq_zero C A).2 h₂]
-      apply flip_collinear_snd_trd
-      exact triv_collinear _ _
+    · simp only [eq_iff_vec_eq_zero.2 h₂]
+      apply Collinear.perm₁₃₂
+      exact triv_collinear₁₂ _ _
     · have g : SameRay ℝ (VEC B C) (VEC C A)
       · rw [sameRay_iff_norm_add, ← eq]
         congr <;>
         exact Seg.length_eq_norm_toVec
       rcases SameRay.exists_pos g h₁ h₂ with ⟨_, ⟨_, ⟨_, ⟨_, g⟩⟩⟩⟩
       rw [← neg_vec C B, ← neg_one_smul ℝ, ← mul_smul, mul_neg_one, ← eq_inv_smul_iff₀ (by linarith), ← mul_smul] at g
-      exact perm_collinear_snd_trd_fst (collinear_of_vec_eq_smul_vec g)
+      exact Collinear.perm₂₃₁ (collinear_of_vec_eq_smul_vec g)
 
 theorem triangle_ineq' (nontriv : tr.IsND) : tr.edge₁.length + tr.edge₂.length > tr.edge₃.length := by
   have ne : tr.edge₁.length + tr.edge₂.length ≠ tr.edge₃.length := by
@@ -345,7 +345,7 @@ theorem nontrivial_of_edge_sum_gt_edge : tr.edge₁.length + tr.edge₂.length >
 
 So funny. Can you get it? -/
 
-theorem edge_sum_eq_edge_iff_collinear : collinear tr.1 tr.2 tr.3 ↔ (tr.edge₁.length + tr.edge₂.length = tr.edge₃.length) ∨ (tr.edge₂.length + tr.edge₃.length = tr.edge₁.length) ∨ (tr.edge₃.length + tr.edge₁.length = tr.edge₂.length) := sorry
+theorem edge_sum_eq_edge_iff_collinear : Collinear tr.1 tr.2 tr.3 ↔ (tr.edge₁.length + tr.edge₂.length = tr.edge₃.length) ∨ (tr.edge₂.length + tr.edge₃.length = tr.edge₁.length) ∨ (tr.edge₃.length + tr.edge₁.length = tr.edge₂.length) := sorry
 /- area ≥ 0, nontrivial → >0, =0 → trivial -/
 
 end Triangle
@@ -374,11 +374,11 @@ theorem angle₁_pos_iff_cclock : tr_nd.is_cclock ↔ tr_nd.angle₁.value.IsPos
     simp only [eq_iff_iff]
     have pos3 : tr_nd.edge₃.length > 0 := by
       calc
-        _= tr_nd.edge_nd₃.length := by rfl
+        _= tr_nd.edgeND₃.length := by rfl
         _>0 := by apply EuclidGeom.length_pos
     have pos2 : tr_nd.edge₂.length > 0 := by
       calc
-        _= tr_nd.edge_nd₂.length := by rfl
+        _= tr_nd.edgeND₂.length := by rfl
         _>0 := by apply EuclidGeom.length_pos
     constructor
     · intro P
diff --git a/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_ex.lean b/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_ex.lean
index 57a41600..29c7c76f 100644
--- a/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_ex.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_ex.lean
@@ -104,15 +104,15 @@ the uniqueness of the graph and how to use it to determine the orientation.
 -/
 variable (tr_nd : TriangleND P)
 
-theorem iscclock_iff_liesonleft₃ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr_nd.1.3 LiesOnLeft tr_nd.edge_nd₃ := by
+theorem iscclock_iff_liesonleft₃ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr_nd.1.3 LiesOnLeft tr_nd.edgeND₃ := by
   unfold is_cclock
   unfold IsOnLeftSide
-  have h : (0 < tr_nd.oarea) = (0 < odist tr_nd.1.3 tr_nd.edge_nd₃) := by
-    have : EuclidGeom.oarea tr_nd.1.1 tr_nd.1.2 tr_nd.1.3 = tr_nd.edge₃.length * odist tr_nd.1.3 tr_nd.edge_nd₃ /2 := by
+  have h : (0 < tr_nd.oarea) = (0 < odist tr_nd.1.3 tr_nd.edgeND₃) := by
+    have : EuclidGeom.oarea tr_nd.1.1 tr_nd.1.2 tr_nd.1.3 = tr_nd.edge₃.length * odist tr_nd.1.3 tr_nd.edgeND₃ /2 := by
       apply oarea_eq_length_mul_odist_div_two
     unfold oarea
     unfold Triangle.oarea
-    have pos : tr_nd.edge_nd₃.length > 0 := by
+    have pos : tr_nd.edgeND₃.length > 0 := by
       apply EuclidGeom.length_pos
     simp only [this, eq_iff_iff]
     symm
@@ -121,17 +121,17 @@ theorem iscclock_iff_liesonleft₃ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr
       positivity
     · intro p
       apply pos_of_mul_pos_left (b := tr_nd.edge₃.length/2)
-      · have simp : odist tr_nd.1.3 tr_nd.edge_nd₃ * (tr_nd.edge₃.length / 2) = tr_nd.edge₃.length * odist tr_nd.1.3 tr_nd.edge_nd₃ / 2  := by ring
+      · have simp : odist tr_nd.1.3 tr_nd.edgeND₃ * (tr_nd.edge₃.length / 2) = tr_nd.edge₃.length * odist tr_nd.1.3 tr_nd.edgeND₃ / 2  := by ring
         rw [simp]
         exact p
       · positivity
   exact h
-theorem iscclock_iff_liesonleft₁ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr_nd.1.1 LiesOnLeft tr_nd.edge_nd₁ := by
+theorem iscclock_iff_liesonleft₁ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr_nd.1.1 LiesOnLeft tr_nd.edgeND₁ := by
   have h : tr_nd.is_cclock = (tr_nd.perm_vertices.is_cclock) := by
     apply same_orient_of_perm_vertices
   simp only [h]
   apply iscclock_iff_liesonleft₃
-theorem iscclock_iff_liesonleft₂ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr_nd.1.2 LiesOnLeft tr_nd.edge_nd₂ := by
+theorem iscclock_iff_liesonleft₂ (tr_nd : TriangleND P) : tr_nd.is_cclock = tr_nd.1.2 LiesOnLeft tr_nd.edgeND₂ := by
   have h : tr_nd.is_cclock = (tr_nd.perm_vertices.is_cclock) := by
     apply same_orient_of_perm_vertices
   simp only [h]
@@ -175,7 +175,7 @@ theorem anti_cclock_of_IsOnOppositeSide (A B C D : P) [hne : PtNe B A] (h : IsOn
 
 lemma liesonleft_ne_pts {A B C : P} [hne : PtNe B A] (h : C LiesOnLeft (DLIN A B)) : (C ≠ A) ∧ (C ≠ B) := by
   have h': C LiesOnLeft (RAY A B) := by exact h
-  have : ¬ collinear A B C := by
+  have : ¬ Collinear A B C := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     simp only [h', true_or]
   have c_ne_a : C ≠ A := (ne_of_not_collinear this).2.1.symm
@@ -184,7 +184,7 @@ lemma liesonleft_ne_pts {A B C : P} [hne : PtNe B A] (h : C LiesOnLeft (DLIN A B
 
 theorem liesonleft_angle_ispos {A B C : P} [hne : PtNe B A] (h : C LiesOnLeft (DLIN A B)) : (∠ A C B (liesonleft_ne_pts h).1.symm (liesonleft_ne_pts h).2.symm).IsPos := by
   have h': C LiesOnLeft (RAY A B) := by exact h
-  have ABC_nd: ¬ collinear A B C := by
+  have ABC_nd: ¬ Collinear A B C := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     simp only [h', true_or]
   have c : (TRI_nd A B C ABC_nd).is_cclock = C LiesOnLeft (SEG_nd A B) := by
@@ -201,7 +201,7 @@ theorem liesonleft_angle_ispos {A B C : P} [hne : PtNe B A] (h : C LiesOnLeft (D
 
 lemma liesonright_ne_pts {A B C : P} [hne : PtNe B A] (h : C LiesOnRight (DLIN A B)) : (C ≠ A) ∧ (C ≠ B) := by
   have h': C LiesOnRight (RAY A B) := by exact h
-  have : ¬ collinear A B C := by
+  have : ¬ Collinear A B C := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     simp only [h', or_true]
   have c_ne_a : C ≠ A := (ne_of_not_collinear this).2.1.symm
@@ -211,7 +211,7 @@ lemma liesonright_ne_pts {A B C : P} [hne : PtNe B A] (h : C LiesOnRight (DLIN A
 theorem liesonright_angle_isneg {A B C : P} [hne : PtNe B A] (h : C LiesOnRight (DLIN A B)) : (∠ A C B (liesonright_ne_pts h).1.symm (liesonright_ne_pts h).2.symm).IsNeg := by
   have h': C LiesOnRight (RAY A B) := by exact h
   have h'' : C LiesOnRight (SEG_nd A B) := by exact h
-  have ABC_nd: ¬ collinear A B C := by
+  have ABC_nd: ¬ Collinear A B C := by
     apply not_collinear_of_LiesOnLeft_or_LiesOnRight
     simp only [h', or_true]
   have c : (TRI_nd A B C ABC_nd).is_cclock = C LiesOnLeft (SEG_nd A B) := by
diff --git a/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_trash.lean b/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_trash.lean
index 001c308c..d546e59d 100644
--- a/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_trash.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Triangle/Basic_trash.lean
@@ -13,7 +13,7 @@ structure TriangleND.IsAcute (tr_nd : TriangleND P) : Prop where
 
 variable {tr_nd₁ tr_nd₂ : TriangleND P}
 
-theorem edge_toLine_not_para_of_not_collinear {A B C : P} (h : ¬ collinear A B C) : ¬ (SEG_nd A B (ne_of_not_collinear h).2.2) ∥ SEG_nd B C (ne_of_not_collinear h).1 ∧ ¬  (SEG_nd B C (ne_of_not_collinear h).1) ∥ SEG_nd C A (ne_of_not_collinear h).2.1 ∧ ¬  (SEG_nd C A (ne_of_not_collinear h).2.1) ∥ SEG_nd A B (ne_of_not_collinear h).2.2 := by
+theorem edge_toLine_not_para_of_not_collinear {A B C : P} (h : ¬ Collinear A B C) : ¬ (SEG_nd A B (ne_of_not_collinear h).2.2) ∥ SEG_nd B C (ne_of_not_collinear h).1 ∧ ¬  (SEG_nd B C (ne_of_not_collinear h).1) ∥ SEG_nd C A (ne_of_not_collinear h).2.1 ∧ ¬  (SEG_nd C A (ne_of_not_collinear h).2.1) ∥ SEG_nd A B (ne_of_not_collinear h).2.2 := by
   constructor
   by_contra h1
   have eq1 : LIN A B (ne_of_not_collinear h).2.2 = LIN B C (ne_of_not_collinear h).1 := by
@@ -22,7 +22,7 @@ theorem edge_toLine_not_para_of_not_collinear {A B C : P} (h : ¬ collinear A B
     exact (SEG_nd B C (ne_of_not_collinear h).1).source_lies_on_toLine
     exact h1
   have a_lies_on_ab : A LiesOn (LIN A B (ne_of_not_collinear h).2.2) := (SEG_nd A B (ne_of_not_collinear h).2.2).source_lies_on_toLine
-  have a_not_lies_on_bc := (Line.lies_on_line_of_pt_pt_iff_collinear (_h := ⟨(ne_of_not_collinear h).1⟩) A).mp.mt (flip_collinear_snd_trd.mt (flip_collinear_fst_snd.mt h))
+  have a_not_lies_on_bc := (Line.lies_on_line_of_pt_pt_iff_collinear (_h := ⟨(ne_of_not_collinear h).1⟩) A).mp.mt (Collinear.perm₁₃₂.mt (Collinear.perm₂₁₃.mt h))
   simp only[← eq1] at a_not_lies_on_bc
   apply a_not_lies_on_bc
   exact a_lies_on_ab
@@ -34,7 +34,7 @@ theorem edge_toLine_not_para_of_not_collinear {A B C : P} (h : ¬ collinear A B
     exact (SEG_nd C A (ne_of_not_collinear h).2.1).source_lies_on_toLine
     exact h2
   have b_lies_on_bc : B LiesOn (LIN B C (ne_of_not_collinear h).1) := (SEG_nd B C (ne_of_not_collinear h).1).source_lies_on_toLine
-  have b_not_lies_on_ca := (Line.lies_on_line_of_pt_pt_iff_collinear (_h := ⟨(ne_of_not_collinear h).2.1⟩) B).mp.mt (flip_collinear_fst_snd.mt (flip_collinear_snd_trd.mt h))
+  have b_not_lies_on_ca := (Line.lies_on_line_of_pt_pt_iff_collinear (_h := ⟨(ne_of_not_collinear h).2.1⟩) B).mp.mt (Collinear.perm₂₁₃.mt (Collinear.perm₁₃₂.mt h))
   simp only[← eq2] at b_not_lies_on_ca
   apply b_not_lies_on_ca
   exact b_lies_on_bc
@@ -71,15 +71,15 @@ theorem edge_eq_edge_of_perm_vertices_two_times (tr_nd : TriangleND P) : tr_nd.1
 
 theorem angle_eq_angle_of_perm_vertices_two_times (tr_nd : TriangleND P) : tr_nd.angle₁.value = tr_nd.perm_vertices.perm_vertices.angle₃.value ∧ tr_nd.angle₂.value = tr_nd.perm_vertices.perm_vertices.angle₁.value ∧ tr_nd.angle₃.value = tr_nd.perm_vertices.perm_vertices.angle₂.value := by sorry
 
-theorem points_ne_of_collinear_of_not_collinear1 {A B C D : P} (ncolin : ¬ collinear A B C) (colin : collinear D B C) : D ≠ A := sorry
+theorem points_ne_of_collinear_of_not_collinear1 {A B C D : P} (ncolin : ¬ Collinear A B C) (colin : Collinear D B C) : D ≠ A := sorry
 
-theorem points_ne_of_collinear_of_not_collinear2 {A B C D : P} (ncolin : ¬ collinear A B C) (colin : collinear D C A) : D ≠ B := sorry
+theorem points_ne_of_collinear_of_not_collinear2 {A B C D : P} (ncolin : ¬ Collinear A B C) (colin : Collinear D C A) : D ≠ B := sorry
 
-theorem points_ne_of_collinear_of_not_collinear3 {A B C D : P} (ncolin : ¬ collinear A B C) (colin : collinear D A B) : D ≠ C := sorry
+theorem points_ne_of_collinear_of_not_collinear3 {A B C D : P} (ncolin : ¬ Collinear A B C) (colin : Collinear D A B) : D ≠ C := sorry
 
-theorem not_parallel_of_not_collinear_of_collinear_collinear {A B C D E : P} (nd : ¬ collinear A B C) (colindbc : collinear D B C) (colineca : collinear E C A) : ¬ (LIN A D (points_ne_of_collinear_of_not_collinear1 nd colindbc)) ∥ (LIN B E (points_ne_of_collinear_of_not_collinear2 nd colineca)) := sorry
+theorem not_parallel_of_not_collinear_of_collinear_collinear {A B C D E : P} (nd : ¬ Collinear A B C) (colindbc : Collinear D B C) (colineca : Collinear E C A) : ¬ (LIN A D (points_ne_of_collinear_of_not_collinear1 nd colindbc)) ∥ (LIN B E (points_ne_of_collinear_of_not_collinear2 nd colineca)) := sorry
 
-theorem intersection_not_collinear_of_nondegenerate {A B C D E : P} (nd : ¬ collinear A B C) (colindbc : collinear D B C) (colineca : collinear E C A) (dneb : D ≠ B) (dnec : D ≠ C) (enea : E ≠ A) (enec : E ≠ C) (F : P) (fdef : F = Line.inx (LIN A D (points_ne_of_collinear_of_not_collinear1 nd colindbc)) (LIN B E (points_ne_of_collinear_of_not_collinear2 nd colineca)) (not_parallel_of_not_collinear_of_collinear_collinear nd colindbc colineca)) : (¬ collinear A B F) ∧ (¬ collinear B C F) ∧ (¬ collinear C A F) := by sorry
+theorem intersection_not_collinear_of_nondegenerate {A B C D E : P} (nd : ¬ Collinear A B C) (colindbc : Collinear D B C) (colineca : Collinear E C A) (dneb : D ≠ B) (dnec : D ≠ C) (enea : E ≠ A) (enec : E ≠ C) (F : P) (fdef : F = Line.inx (LIN A D (points_ne_of_collinear_of_not_collinear1 nd colindbc)) (LIN B E (points_ne_of_collinear_of_not_collinear2 nd colineca)) (not_parallel_of_not_collinear_of_collinear_collinear nd colindbc colineca)) : (¬ Collinear A B F) ∧ (¬ Collinear B C F) ∧ (¬ Collinear C A F) := by sorry
 
 theorem cclock_of_eq_angle (tr_nd₁ tr_nd₂ : TriangleND P)(a : tr_nd₁.angle₁.value = tr_nd₂.angle₁.value) : tr_nd₁.is_cclock ↔ tr_nd₂.is_cclock := by sorry
 
diff --git a/EuclideanGeometry/Foundation/Axiom/Triangle/Congruence.lean b/EuclideanGeometry/Foundation/Axiom/Triangle/Congruence.lean
index d14743d2..d80a840e 100644
--- a/EuclideanGeometry/Foundation/Axiom/Triangle/Congruence.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Triangle/Congruence.lean
@@ -41,17 +41,17 @@ theorem nd_of_nd (h : tr₁.IsCongr tr₂) (nd : tr₁.IsND) : tr₂.IsND := by
   rw [Triangle.edge_sum_eq_edge_iff_collinear] at col
   rcases col with l₁ | l₂ | l₃
   . simp only [<-h.1, <-h.2, <-h.3] at l₁
-    have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+    have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
       rw [Triangle.edge_sum_eq_edge_iff_collinear]
       exact .inl l₁
     exact nd col'
   . simp only [<-h.1, <-h.2, <-h.3] at l₂
-    have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+    have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
       rw [Triangle.edge_sum_eq_edge_iff_collinear]
       exact .inr (.inl l₂)
     exact nd col'
   . simp only [<-h.1, <-h.2, <-h.3] at l₃
-    have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+    have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
       rw [Triangle.edge_sum_eq_edge_iff_collinear]
       exact .inr (.inr l₃)
     exact nd col'
@@ -161,17 +161,17 @@ theorem nd_of_nd (h : tr₁.IsACongr tr₂) (nd : tr₁.IsND) : tr₂.IsND := by
   rw [Triangle.edge_sum_eq_edge_iff_collinear] at col
   rcases col with l₁ | l₂ | l₃
   . simp only [<-h.1, <-h.2, <-h.3] at l₁
-    have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+    have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
       rw [Triangle.edge_sum_eq_edge_iff_collinear]
       exact .inl l₁
     exact nd col'
   . simp only [<-h.1, <-h.2, <-h.3] at l₂
-    have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+    have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
       rw [Triangle.edge_sum_eq_edge_iff_collinear]
       exact .inr (.inl l₂)
     exact nd col'
   . simp only [<-h.1, <-h.2, <-h.3] at l₃
-    have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+    have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
       rw [Triangle.edge_sum_eq_edge_iff_collinear]
       exact .inr (.inr l₃)
     exact nd col'
@@ -590,9 +590,9 @@ theorem cosine_eq_of_SSS (e₁ : tr_nd₁.1.edge₁.length = tr_nd₂.1.edge₁.
   rw [e₁,e₂,e₃,←cos₂] at cos₁
   field_simp at cos₁
   have u₁ : 0 < tr_nd₂.1.edge₃.length := by
-    exact length_pos_iff_nd.mpr tr_nd₂.edge_nd₃.2
+    exact length_pos_iff_nd.mpr tr_nd₂.edgeND₃.2
   have u₂ : 0 < tr_nd₂.1.edge₂.length := by
-    exact length_pos_iff_nd.mpr tr_nd₂.edge_nd₂.2
+    exact length_pos_iff_nd.mpr tr_nd₂.edgeND₂.2
   have h0 : (tr_nd₂.1.edge₃.length * tr_nd₂.1.edge₂.length) > 0 := by
     field_simp [u₁,u₂]
   rcases cos₁ with x | y
@@ -930,17 +930,17 @@ theorem congr_or_acongr_of_SSS (e₁ : tr₁.edge₁.length = tr₂.edge₁.leng
       rw [Triangle.edge_sum_eq_edge_iff_collinear] at nd₂
       rcases nd₂ with l₁ | l₂ | l₃
       . simp only [<-e₁, <-e₂, <-e₃] at l₁
-        have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+        have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
           rw [Triangle.edge_sum_eq_edge_iff_collinear]
           exact .inl l₁
         exact nd₁ col'
       . simp only [<-e₁, <-e₂, <-e₃] at l₂
-        have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+        have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
           rw [Triangle.edge_sum_eq_edge_iff_collinear]
           exact .inr (.inl l₂)
         exact nd₁ col'
       . simp only [<-e₁, <-e₂, <-e₃] at l₃
-        have col' : collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
+        have col' : Collinear tr₁.point₁ tr₁.point₂ tr₁.point₃ := by
           rw [Triangle.edge_sum_eq_edge_iff_collinear]
           exact .inr (.inr l₃)
         exact nd₁ col'
diff --git a/EuclideanGeometry/Foundation/Axiom/Triangle/Similarity.lean b/EuclideanGeometry/Foundation/Axiom/Triangle/Similarity.lean
index 3eac3dbc..18e345b5 100644
--- a/EuclideanGeometry/Foundation/Axiom/Triangle/Similarity.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Triangle/Similarity.lean
@@ -70,17 +70,17 @@ theorem ratio₂ : h.ratio = tr₁.edge₂.length / tr₂.edge₂.length := by
   rw [h.1,h.2] at sine₁
   have s₁₂: tr₁.edge₁.length / tr₁.edge₂.length = (sin (Angle.value tr₂.angle₁)) / (sin (Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₁
-    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₁.edge_nd₂.2).symm).mpr sine₁).symm
+    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₁.edgeND₂.2).symm).mpr sine₁).symm
   have t₁₂: tr₂.edge₁.length / tr₂.edge₂.length = (sin (Angle.value tr₂.angle₁)) / (sin (Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₂
-    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mpr sine₂).symm
+    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mpr sine₂).symm
   rw [<-t₁₂] at s₁₂
   unfold ratio
-  have eq := mul_eq_mul_of_div_eq_div tr₁.edge₁.length tr₂.edge₁.length (length_ne_zero_iff_nd.mpr tr₁.edge_nd₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm s₁₂
-  apply (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edge_nd₁.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mpr
-  have e₂ : (tr₂.edge_nd₂).1.length = tr₂.edge₂.length := rfl
-  have e₁ : (tr₂.edge_nd₁).1.length = tr₂.edge₁.length := rfl
-  have e₂' : (tr₁.edge_nd₂).1.length = tr₁.edge₂.length := rfl
+  have eq := mul_eq_mul_of_div_eq_div tr₁.edge₁.length tr₂.edge₁.length (length_ne_zero_iff_nd.mpr tr₁.edgeND₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm s₁₂
+  apply (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edgeND₁.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mpr
+  have e₂ : (tr₂.edgeND₂).1.length = tr₂.edge₂.length := rfl
+  have e₁ : (tr₂.edgeND₁).1.length = tr₂.edge₁.length := rfl
+  have e₂' : (tr₁.edgeND₂).1.length = tr₁.edge₂.length := rfl
   rw [e₂, e₂'] at eq
   simp only [e₂, e₁, eq, mul_comm]
 
@@ -93,10 +93,10 @@ theorem ratio₁₂ : tr₁.edge₁.length / tr₁.edge₂.length = tr₂.edge
   rw [h.1,h.2] at sine₁
   have s₁₂: tr₁.edge₁.length / tr₁.edge₂.length = (sin (Angle.value tr₂.angle₁)) / (sin (Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₁
-    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₁.edge_nd₂.2).symm).mpr sine₁).symm
+    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₁.edgeND₂.2).symm).mpr sine₁).symm
   have t₁₂: tr₂.edge₁.length / tr₂.edge₂.length = (sin (Angle.value tr₂.angle₁)) / (sin (Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₂
-    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mpr sine₂).symm
+    exact ((div_eq_div_iff  (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mpr sine₂).symm
   rw [<-t₁₂] at s₁₂
   exact s₁₂
 
@@ -155,18 +155,18 @@ theorem ratio₂ : h.ratio = tr₁.edge₂.length / tr₂.edge₂.length := by
   rw [<-this,<-(angle_eq_angle_of_perm_vertices (perm_vertices tr₂)).2.2,<-(angle_eq_angle_of_perm_vertices  tr₂).2.1] at nd
   have s₁₂: tr₁.edge₁.length / tr₁.edge₂.length = (sin (- Angle.value tr₂.angle₁)) / (sin (-Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₁
-    exact ((div_eq_div_iff nd (length_ne_zero_iff_nd.mpr tr₁.edge_nd₂.2).symm).mpr sine₁).symm
+    exact ((div_eq_div_iff nd (length_ne_zero_iff_nd.mpr tr₁.edgeND₂.2).symm).mpr sine₁).symm
   have t₁₂: tr₂.edge₁.length / tr₂.edge₂.length = (sin (Angle.value tr₂.angle₁)) / (sin (Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₂
-    exact ((div_eq_div_iff (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mpr sine₂).symm
+    exact ((div_eq_div_iff (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mpr sine₂).symm
   rw [sin_neg (Angle.value tr₂.angle₁),sin_neg (Angle.value tr₂.angle₂),neg_div_neg_eq] at s₁₂
   rw [<-t₁₂] at s₁₂
   unfold ratio
-  have eq := mul_eq_mul_of_div_eq_div tr₁.edge₁.length tr₂.edge₁.length (length_ne_zero_iff_nd.mpr tr₁.edge_nd₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm s₁₂
-  apply (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edge_nd₁.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mpr
-  have e₂ : (tr₂.edge_nd₂).1.length = tr₂.edge₂.length := rfl
-  have e₁ : (tr₂.edge_nd₁).1.length = tr₂.edge₁.length := rfl
-  have e₂' : (tr₁.edge_nd₂).1.length = tr₁.edge₂.length := rfl
+  have eq := mul_eq_mul_of_div_eq_div tr₁.edge₁.length tr₂.edge₁.length (length_ne_zero_iff_nd.mpr tr₁.edgeND₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm s₁₂
+  apply (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edgeND₁.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mpr
+  have e₂ : (tr₂.edgeND₂).1.length = tr₂.edge₂.length := rfl
+  have e₁ : (tr₂.edgeND₁).1.length = tr₂.edge₁.length := rfl
+  have e₂' : (tr₁.edgeND₂).1.length = tr₁.edge₂.length := rfl
   rw [e₂, e₂'] at eq
   simp only [e₂, e₁, eq, mul_comm]
 
@@ -183,10 +183,10 @@ theorem ratio₁₂ : tr₁.edge₁.length / tr₁.edge₂.length = tr₂.edge
   rw [<-this,<-(angle_eq_angle_of_perm_vertices (perm_vertices tr₂)).2.2,<-(angle_eq_angle_of_perm_vertices  tr₂).2.1] at nd
   have s₁₂: tr₁.edge₁.length / tr₁.edge₂.length = (sin (- Angle.value tr₂.angle₁)) / (sin (-Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₁
-    exact ((div_eq_div_iff nd (length_ne_zero_iff_nd.mpr tr₁.edge_nd₂.2).symm).mpr sine₁).symm
+    exact ((div_eq_div_iff nd (length_ne_zero_iff_nd.mpr tr₁.edgeND₂.2).symm).mpr sine₁).symm
   have t₁₂: tr₂.edge₁.length / tr₂.edge₂.length = (sin (Angle.value tr₂.angle₁)) / (sin (Angle.value tr₂.angle₂)) := by
     rw [mul_comm] at sine₂
-    exact ((div_eq_div_iff (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mpr sine₂).symm
+    exact ((div_eq_div_iff (sine_ne_zero_of_nd (perm_vertices tr₂)) (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mpr sine₂).symm
   rw [sin_neg (Angle.value tr₂.angle₁), sin_neg (Angle.value tr₂.angle₂), neg_div_neg_eq, ← t₁₂] at s₁₂
   exact s₁₂
 
@@ -284,9 +284,9 @@ theorem asim_of_AA (tr₁ tr₂ : TriangleND P) (h₂ : tr₁.angle₂.value = -
 /- SAS -/
 theorem sim_of_SAS (tr₁ tr₂ : TriangleND P) (e : tr₁.edge₂.length / tr₂.edge₂.length = tr₁.edge₃.length / tr₂.edge₃.length) (a : tr₁.angle₁.value = tr₂.angle₁.value): tr₁ ∼ tr₂ := by
   have eq : tr₁.edge₂.length * tr₂.edge₃.length = tr₁.edge₃.length * tr₂.edge₂.length := by
-    exact (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm).mp e
+    exact (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm).mp e
   rw [mul_comm tr₁.edge₃.length  tr₂.edge₂.length] at eq
-  have e' : tr₁.edge₂.length / tr₁.edge₃.length = tr₂.edge₂.length / tr₂.edge₃.length := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edge_nd₃.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm).mpr eq
+  have e' : tr₁.edge₂.length / tr₁.edge₃.length = tr₂.edge₂.length / tr₂.edge₃.length := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edgeND₃.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm).mpr eq
   have sine₁ := Triangle.sine_rule₁ tr₁
   have sine₂ := Triangle.sine_rule₁ tr₂
   have sine₁' : tr₁.edge₂.length * sin (Angle.value tr₁.angle₃) = sin (Angle.value tr₁.angle₂) *  tr₁.edge₃.length := by
@@ -299,8 +299,8 @@ theorem sim_of_SAS (tr₁ tr₂ : TriangleND P) (e : tr₁.edge₂.length / tr
   have ne₂₃ : sin (Angle.value tr₂.angle₃) ≠ 0 := by
     simp only [(angle_eq_angle_of_perm_vertices tr₂).2.2,ne_eq, sine_ne_zero_of_nd,
         not_false_eq_true]
-  have s₁₂ : tr₁.edge₂.length / tr₁.edge₃.length = sin (Angle.value tr₁.angle₂) / sin (Angle.value tr₁.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edge_nd₃.2).symm ne₁₃).mpr sine₁'
-  have t₁₂ : tr₂.edge₂.length / tr₂.edge₃.length = sin (Angle.value tr₂.angle₂) / sin (Angle.value tr₂.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm ne₂₃).mpr sine₂'
+  have s₁₂ : tr₁.edge₂.length / tr₁.edge₃.length = sin (Angle.value tr₁.angle₂) / sin (Angle.value tr₁.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edgeND₃.2).symm ne₁₃).mpr sine₁'
+  have t₁₂ : tr₂.edge₂.length / tr₂.edge₃.length = sin (Angle.value tr₂.angle₂) / sin (Angle.value tr₂.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm ne₂₃).mpr sine₂'
   rw [s₁₂,t₁₂] at e'
   have e'' := (div_eq_div_iff ne₁₃ ne₂₃).mp e'
   have summul := Real.cos_sub_cos ((Angle.value tr₁.angle₂).toReal + (Angle.value tr₂.angle₃).toReal) ((Angle.value tr₁.angle₂).toReal - (Angle.value tr₂.angle₃).toReal)
@@ -413,9 +413,9 @@ theorem sim_of_SAS (tr₁ tr₂ : TriangleND P) (e : tr₁.edge₂.length / tr
 
 theorem asim_of_SAS (tr₁ tr₂ : TriangleND P) (e : tr₁.edge₂.length / tr₂.edge₂.length = tr₁.edge₃.length / tr₂.edge₃.length) (a : tr₁.angle₁.value = - tr₂.angle₁.value): tr₁ ∼ₐ tr₂ := by
   have eq : tr₁.edge₂.length * tr₂.edge₃.length = tr₁.edge₃.length * tr₂.edge₂.length := by
-    exact (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm).mp e
+    exact (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm).mp e
   rw [mul_comm tr₁.edge₃.length  tr₂.edge₂.length] at eq
-  have e' : tr₁.edge₂.length / tr₁.edge₃.length = tr₂.edge₂.length / tr₂.edge₃.length := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edge_nd₃.2).symm (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm).mpr eq
+  have e' : tr₁.edge₂.length / tr₁.edge₃.length = tr₂.edge₂.length / tr₂.edge₃.length := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edgeND₃.2).symm (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm).mpr eq
   have sine₁ := Triangle.sine_rule₁ tr₁
   have sine₂ := Triangle.sine_rule₁ tr₂
   have sine₁' : tr₁.edge₂.length * sin (Angle.value tr₁.angle₃) = sin (Angle.value tr₁.angle₂) *  tr₁.edge₃.length := by
@@ -426,8 +426,8 @@ theorem asim_of_SAS (tr₁ tr₂ : TriangleND P) (e : tr₁.edge₂.length / tr
     simp only [(angle_eq_angle_of_perm_vertices tr₁).2.2, ne_eq,sine_ne_zero_of_nd, not_false_eq_true]
   have ne₂₃ : sin (Angle.value tr₂.angle₃) ≠ 0 := by
     simp only [(angle_eq_angle_of_perm_vertices tr₂).2.2,ne_eq, sine_ne_zero_of_nd, not_false_eq_true]
-  have s₁₂ : tr₁.edge₂.length / tr₁.edge₃.length = sin (Angle.value tr₁.angle₂) / sin (Angle.value tr₁.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edge_nd₃.2).symm ne₁₃).mpr  sine₁'
-  have t₁₂ : tr₂.edge₂.length / tr₂.edge₃.length = sin (Angle.value tr₂.angle₂) / sin (Angle.value tr₂.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm ne₂₃).mpr  sine₂'
+  have s₁₂ : tr₁.edge₂.length / tr₁.edge₃.length = sin (Angle.value tr₁.angle₂) / sin (Angle.value tr₁.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₁.edgeND₃.2).symm ne₁₃).mpr  sine₁'
+  have t₁₂ : tr₂.edge₂.length / tr₂.edge₃.length = sin (Angle.value tr₂.angle₂) / sin (Angle.value tr₂.angle₃) := (div_eq_div_iff (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm ne₂₃).mpr  sine₂'
   rw [s₁₂,t₁₂] at e'
   have e'' := (div_eq_div_iff ne₁₃ ne₂₃).mp e'
   have summul := Real.cos_sub_cos ((Angle.value tr₁.angle₂).toReal + (Angle.value tr₂.angle₃).toReal) ((Angle.value tr₁.angle₂).toReal - (Angle.value tr₂.angle₃).toReal)
@@ -563,13 +563,13 @@ theorem IsSim.congr_of_ratio_eq_one (h : IsSim tr₁ tr₂) (hr : h.ratio = 1) :
   constructor
   have := ratio₁ h
   rw [hr] at this
-  exact (div_eq_one_iff_eq (length_ne_zero_iff_nd.mpr tr₂.edge_nd₁.2).symm).mp this.symm
+  exact (div_eq_one_iff_eq (length_ne_zero_iff_nd.mpr tr₂.edgeND₁.2).symm).mp this.symm
   have := ratio₂ h
   rw [hr] at this
-  exact (div_eq_one_iff_eq (length_ne_zero_iff_nd.mpr tr₂.edge_nd₂.2).symm).mp this.symm
+  exact (div_eq_one_iff_eq (length_ne_zero_iff_nd.mpr tr₂.edgeND₂.2).symm).mp this.symm
   have := ratio₃ h
   rw [hr] at this
-  exact (div_eq_one_iff_eq (length_ne_zero_iff_nd.mpr tr₂.edge_nd₃.2).symm).mp this.symm
+  exact (div_eq_one_iff_eq (length_ne_zero_iff_nd.mpr tr₂.edgeND₃.2).symm).mp this.symm
   exact h.1
   exact h.2
   exact h.3
diff --git a/EuclideanGeometry/Foundation/Axiom/Triangle/Trigonometric.lean b/EuclideanGeometry/Foundation/Axiom/Triangle/Trigonometric.lean
index 6a224272..96c8ae64 100644
--- a/EuclideanGeometry/Foundation/Axiom/Triangle/Trigonometric.lean
+++ b/EuclideanGeometry/Foundation/Axiom/Triangle/Trigonometric.lean
@@ -70,7 +70,7 @@ theorem Pythagoras_of_perp_foot (A B : P) {l : Line P} (h : B LiesOn l) : (SEG A
   --let C := tr_nd.1.point₃
 
 /-- Given ▵ A B C with ∠ B A C = π / 2, A B ^ 2 + A C ^ 2 = B C ^ 2, namely (SEG A B).length ^ 2 + (SEG A C).length ^ 2 = (SEG B C).length ^ 2. -/
-theorem Pythagoras_of_right_triangle_non_trivial (A B C : P) {hnd : ¬ collinear A B C} (right_triangle: ∠ B A C (ne_of_not_collinear hnd).2.2 (ne_of_not_collinear hnd).2.1.symm = (π / 2 : ℝ)) : (SEG A B).length ^ 2 + (SEG A C).length ^ 2 = (SEG B C).length ^ 2 := by
+theorem Pythagoras_of_right_triangle_non_trivial (A B C : P) {hnd : ¬ Collinear A B C} (right_triangle: ∠ B A C (ne_of_not_collinear hnd).2.2 (ne_of_not_collinear hnd).2.1.symm = (π / 2 : ℝ)) : (SEG A B).length ^ 2 + (SEG A C).length ^ 2 = (SEG B C).length ^ 2 := by
   have h : cos (∠ B A C (ne_of_not_collinear hnd).2.2 (ne_of_not_collinear hnd).2.1.symm) = 0
   · rw [right_triangle]
     simp
diff --git a/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva.lean b/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva.lean
index 19d384a9..e817fec9 100644
--- a/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva.lean
+++ b/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva.lean
@@ -14,9 +14,9 @@ class CevaCfgClass (P : outParam <| Type*) [outParam <| EuclideanPlane P] where
   (A B C : P)
   --Let $D$ be a point not lying on $AB$, $BC$ and $CA$.
   D : P
-  abd_nd : ¬collinear A B D
-  bcd_nd : ¬collinear B C D
-  cad_nd : ¬collinear C A D
+  abd_nd : ¬Collinear A B D
+  bcd_nd : ¬Collinear B C D
+  cad_nd : ¬Collinear C A D
 
   a_ne_b : PtNe A B := ⟨(ne_of_not_collinear abd_nd).2.2.symm⟩
   b_ne_c : PtNe B C := ⟨(ne_of_not_collinear bcd_nd).2.2.symm⟩
@@ -47,90 +47,90 @@ variable {P : Type*} [EuclideanPlane P] [cfg : CevaCfgClass P]
 attribute [instance] a_ne_b b_ne_c c_ne_a d_ne_a d_ne_b d_ne_c
 
 --$D,C,A$ are not collinear
-lemma ncolin_dca : ¬ collinear D C A := by
+lemma ncolin_dca : ¬ Collinear D C A := by
   intro h0
-  exact cad_nd (perm_collinear_snd_trd_fst h0)
+  exact cad_nd (Collinear.perm₂₃₁ h0)
 
 --$E,B,C$ are collinear
-lemma colin_ebc : collinear E B C := by
+lemma colin_ebc : Collinear E B C := by
   have h : E LiesOn LIN C B := by
     rw [e_def]
     apply Line.inx_lies_on_fst
-  exact flip_collinear_fst_trd (Line.pt_pt_linear h)
+  exact Collinear.perm₃₂₁ (Line.pt_pt_linear h)
 
 --$E,D,A$ are collinear
-lemma colin_eda : collinear E D A := by
+lemma colin_eda : Collinear E D A := by
   have h : E LiesOn LIN A D := by
     rw [e_def]
     apply Line.inx_lies_on_snd
-  exact flip_collinear_fst_trd (Line.pt_pt_linear h)
+  exact Collinear.perm₃₂₁ (Line.pt_pt_linear h)
 
 --$C\ne E$
 instance c_ne_e : PtNe C E := Fact.mk <| by
-  have h : collinear E D A := colin_eda
+  have h : Collinear E D A := colin_eda
   intro k
   rw [←k] at h
-  exact ncolin_dca (flip_collinear_fst_snd h)
+  exact ncolin_dca (Collinear.perm₂₁₃ h)
 
 --$EB/EC=S_{\trian}DBA/S_{\trian}DCA$
 lemma dratio_ebc_eq_wedge_div_wedge : divratio E B C = (wedge D B A) / (wedge D C A) :=
   ratio_eq_wedge_div_wedge_of_collinear_collinear_notcoliear E B C D A colin_ebc colin_eda ncolin_dca
 
 --$D,A,B$ are not collinear
-lemma ncolin_dab : ¬ collinear D A B := by
+lemma ncolin_dab : ¬ Collinear D A B := by
   intro h0
-  exact abd_nd (perm_collinear_snd_trd_fst h0)
+  exact abd_nd (Collinear.perm₂₃₁ h0)
 
 --$F,C,A$ are collinear
-lemma colin_fca : collinear F C A := by
+lemma colin_fca : Collinear F C A := by
   have h : F LiesOn LIN A C := by
     rw [f_def]
     apply Line.inx_lies_on_fst
-  exact flip_collinear_fst_trd (Line.pt_pt_linear h)
+  exact Collinear.perm₃₂₁ (Line.pt_pt_linear h)
 
 --$F,D,B$ are collinear
-lemma colin_fdb : collinear F D B := by
+lemma colin_fdb : Collinear F D B := by
   have h : F LiesOn LIN B D := by
     rw [f_def]
     apply Line.inx_lies_on_snd
-  exact flip_collinear_fst_trd (Line.pt_pt_linear h)
+  exact Collinear.perm₃₂₁ (Line.pt_pt_linear h)
 
 --$A\ne F$
 instance a_ne_f : PtNe A F := Fact.mk <| by
-  have h : collinear F D B := colin_fdb
+  have h : Collinear F D B := colin_fdb
   intro k
   rw [←k] at h
-  exact ncolin_dab (flip_collinear_fst_snd h)
+  exact ncolin_dab (Collinear.perm₂₁₃ h)
 
 --$FC/FA=S_{\trian}DCB/S_{\trian}DAB$
 lemma dratio_fca_eq_wedge_div_wedge : divratio F C A = (wedge D C B) / (wedge D A B) :=
   ratio_eq_wedge_div_wedge_of_collinear_collinear_notcoliear F C A D B colin_fca colin_fdb ncolin_dab
 
 --$D,B,C$ are not collinear
-lemma ncolin_dbc : ¬ collinear D B C := by
+lemma ncolin_dbc : ¬ Collinear D B C := by
   intro h0
-  exact bcd_nd (perm_collinear_snd_trd_fst h0)
+  exact bcd_nd (Collinear.perm₂₃₁ h0)
 
 --$G,A,B$ are collinear
-lemma colin_gab : collinear G A B := by
+lemma colin_gab : Collinear G A B := by
   have h : G LiesOn LIN B A _ := by
     rw [g_def]
     apply Line.inx_lies_on_fst
-  exact flip_collinear_fst_trd (Line.pt_pt_linear h)
+  exact Collinear.perm₃₂₁ (Line.pt_pt_linear h)
 
 --$G,D,C$ are collinear
-lemma colin_gdc : collinear G D C := by
+lemma colin_gdc : Collinear G D C := by
   have h : G LiesOn LIN C D _ := by
     rw [g_def]
     apply Line.inx_lies_on_snd
-  exact flip_collinear_fst_trd (Line.pt_pt_linear h)
+  exact Collinear.perm₃₂₁ (Line.pt_pt_linear h)
 
 --$A\ne F$
 instance b_ne_g : PtNe B G := Fact.mk <| by
-  have h : collinear G D C := colin_gdc
+  have h : Collinear G D C := colin_gdc
   intro k
   rw [←k] at h
-  exact ncolin_dbc (flip_collinear_fst_snd h)
+  exact ncolin_dbc (Collinear.perm₂₁₃ h)
 
 --$GA/GB=S_{\trian}DAC/S_{\trian}DBC$
 lemma dratio_gab_eq_wedge_div_wedge : divratio G A B = (wedge D A C) / (wedge D B C) := ratio_eq_wedge_div_wedge_of_collinear_collinear_notcoliear G A B D C colin_gab colin_gdc ncolin_dbc
@@ -155,9 +155,9 @@ lemma CevaCfg.ceva_theorem {P : Type*} [EuclideanPlane P] (cfg : CevaCfg P) :
   CevaCfgClass.ceva_theorem (cfg := cfg)
 
 variable {P : Type*} [EuclideanPlane P] {A B C D : P}
-  (abd_nd : ¬collinear A B D)
-  (bcd_nd : ¬collinear B C D)
-  (cad_nd : ¬collinear C A D)
+  (abd_nd : ¬Collinear A B D)
+  (bcd_nd : ¬Collinear B C D)
+  (cad_nd : ¬Collinear C A D)
   (ba_npara_cd : ¬ LIN B A (ne_of_not_collinear abd_nd).2.2.symm ∥ LIN C D (ne_of_not_collinear cad_nd).2.1.symm)
   (cb_npara_ad : ¬ LIN C B (ne_of_not_collinear bcd_nd).2.2.symm ∥ LIN A D (ne_of_not_collinear abd_nd).2.1.symm)
   (ac_npara_bd : ¬ LIN A C (ne_of_not_collinear cad_nd).2.2.symm ∥ LIN B D (ne_of_not_collinear bcd_nd).2.1.symm)
diff --git a/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva_converse.lean b/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva_converse.lean
index c9c5ff53..5b6da2ab 100644
--- a/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva_converse.lean
+++ b/EuclideanGeometry/Foundation/Construction/ComputationTool/Ceva_converse.lean
@@ -13,16 +13,16 @@ structure Setting (P : Type _) [EuclideanPlane P] where
   A : P
   B : P
   C : P
-  not_collinear_ABC : ¬ collinear A B C
+  not_collinear_ABC : ¬ Collinear A B C
   -- Let $E$ be a point on $BC$.
   E : P
-  collinear_EBC : collinear E B C
+  collinear_EBC : Collinear E B C
   -- Let $F$ be a point on $CA$.
   F : P
-  collinear_FCA : collinear F C A
+  collinear_FCA : Collinear F C A
   -- Let $G$ be a point on $AB$.
   G : P
-  collinear_GAB : collinear G A B
+  collinear_GAB : Collinear G A B
   -- Claim : $E \ne A$.
   E_ne_A : E ≠ A := TriangleND.points_ne_of_collinear_of_not_collinear1 not_collinear_ABC collinear_EBC
   -- Claim : $F \ne B$.
@@ -64,11 +64,11 @@ theorem Ceva_converse_theorem {P : Type _} [EuclideanPlane P] (e : Setting P) :
     rw [h1] at ratiomul
     field_simp at ratiomul
   /-
-  have collinear_DAE : collinear e.A e.E D := Line.pt_pt_linear (TriangleND.points_ne_of_collinear_of_not_collinear1 e.not_collinear_ABC e.collinear_EBC) (Line.inx_lies_on_fst (TriangleND.not_parallel_of_not_collinear_of_collinear_collinear e.not_collinear_ABC e.collinear_EBC e.collinear_FCA))
+  have collinear_DAE : Collinear e.A e.E D := Line.pt_pt_linear (TriangleND.points_ne_of_collinear_of_not_collinear1 e.not_collinear_ABC e.collinear_EBC) (Line.inx_lies_on_fst (TriangleND.not_parallel_of_not_collinear_of_collinear_collinear e.not_collinear_ABC e.collinear_EBC e.collinear_FCA))
   -/
-  have abd_nd : ¬ collinear e.A e.B D := (TriangleND.intersection_not_collinear_of_nondegenerate e.not_collinear_ABC e.collinear_EBC e.collinear_FCA E_ne_B E_ne_C F_ne_A F_ne_C D rfl).1
-  have bcd_nd : ¬ collinear e.B e.C D := (TriangleND.intersection_not_collinear_of_nondegenerate e.not_collinear_ABC e.collinear_EBC e.collinear_FCA E_ne_B E_ne_C F_ne_A F_ne_C D rfl).2.1
-  have cad_nd : ¬ collinear e.C e.A D := (TriangleND.intersection_not_collinear_of_nondegenerate e.not_collinear_ABC e.collinear_EBC e.collinear_FCA E_ne_B E_ne_C F_ne_A F_ne_C D rfl).2.2
+  have abd_nd : ¬ Collinear e.A e.B D := (TriangleND.intersection_not_collinear_of_nondegenerate e.not_collinear_ABC e.collinear_EBC e.collinear_FCA E_ne_B E_ne_C F_ne_A F_ne_C D rfl).1
+  have bcd_nd : ¬ Collinear e.B e.C D := (TriangleND.intersection_not_collinear_of_nondegenerate e.not_collinear_ABC e.collinear_EBC e.collinear_FCA E_ne_B E_ne_C F_ne_A F_ne_C D rfl).2.1
+  have cad_nd : ¬ Collinear e.C e.A D := (TriangleND.intersection_not_collinear_of_nondegenerate e.not_collinear_ABC e.collinear_EBC e.collinear_FCA E_ne_B E_ne_C F_ne_A F_ne_C D rfl).2.2
   sorry
 
 end Ceva's_converse_theorem
diff --git a/EuclideanGeometry/Foundation/Construction/ComputationTool/Oarea_method.lean b/EuclideanGeometry/Foundation/Construction/ComputationTool/Oarea_method.lean
index f838cc2c..40a3ee28 100644
--- a/EuclideanGeometry/Foundation/Construction/ComputationTool/Oarea_method.lean
+++ b/EuclideanGeometry/Foundation/Construction/ComputationTool/Oarea_method.lean
@@ -18,16 +18,16 @@ theorem wedge_add_wedge_eq_wedge_add_wedge (A B C D : P) : wedge A B C + wedge C
   rw [← Vec.det_skew (VEC A C) (VEC A B),← Vec.det_skew (VEC A D) (VEC A B), ← Vec.det_skew (VEC A D) (VEC A C)]
   ring
 
-theorem wedge_eq_wedge_add_wedge_of_collinear (A B C D : P) (colin : collinear A B C) : wedge D A C = wedge D B C + wedge D A B := by
-  rw[← wedge231 D B C, ← wedge_add_wedge_eq_wedge_add_wedge,← wedge231 C D A, (collinear_iff_wedge_eq_zero A B C).mp colin]
+theorem wedge_eq_wedge_add_wedge_of_collinear (A B C D : P) (colin : Collinear A B C) : wedge D A C = wedge D B C + wedge D A B := by
+  rw[← wedge312 D B C, ← wedge_add_wedge_eq_wedge_add_wedge,← wedge312 C D A, (collinear_iff_wedge_eq_zero A B C).mp colin]
   field_simp
 
-theorem wedge_eq_divratio_mul_wedge_of_collinear (A B C D : P) (colin : collinear A B C) [cnea : PtNe C A] : wedge A B D = (divratio A B C) * wedge A C D := by
+theorem wedge_eq_divratio_mul_wedge_of_collinear (A B C D : P) (colin : Collinear A B C) [cnea : PtNe C A] : wedge A B D = (divratio A B C) * wedge A C D := by
   unfold wedge
   rw [vec_eq_vec_smul_ratio A B C colin]
   simp only [map_smul, LinearMap.smul_apply, smul_eq_mul]
 
-theorem odist_eq_divratio_mul_odist (A B C : P) (dl : DirLine P) (colin : collinear A B C) [cnea : PtNe C A] (aisondl : A LiesOn dl) : odist B dl = (divratio A B C) * odist C dl := by
+theorem odist_eq_divratio_mul_odist (A B C : P) (dl : DirLine P) (colin : Collinear A B C) [cnea : PtNe C A] (aisondl : A LiesOn dl) : odist B dl = (divratio A B C) * odist C dl := by
   have h0 : odist' B (Ray.mk_pt_dirline A dl aisondl)= odist B (Ray.mk_pt_dirline A dl aisondl).toDirLine := rfl
   have h1 : odist B dl = odist' B (Ray.mk_pt_dirline A dl aisondl) := by
     rw[h0,ray_of_pt_dirline_toDirLine_eq_dirline A dl aisondl]
@@ -40,11 +40,11 @@ theorem odist_eq_divratio_mul_odist (A B C : P) (dl : DirLine P) (colin : collin
   rw [h4, vec_eq_vec_smul_ratio A B C colin]
   simp only [map_smul, smul_eq_mul]
 
-theorem  wedge_eq_divratio_mul_wedge_of_collinear_collinear (A B C D E : P) (colin : collinear A B C) [cnea : PtNe C A] (colin' : collinear A D E) : wedge D B E = (divratio A B C) * wedge D C E := by
-  rw[← wedge231 D B E, wedge_eq_wedge_add_wedge_of_collinear E A D B (perm_collinear_trd_fst_snd colin'), ← wedge231 D C E, wedge_eq_wedge_add_wedge_of_collinear E A D C (perm_collinear_trd_fst_snd colin'), wedge213 A B D, wedge231 A B E, wedge213 A C D, wedge231 A C E,wedge_eq_divratio_mul_wedge_of_collinear A B C D colin,wedge_eq_divratio_mul_wedge_of_collinear A B C E colin]
+theorem  wedge_eq_divratio_mul_wedge_of_collinear_collinear (A B C D E : P) (colin : Collinear A B C) [cnea : PtNe C A] (colin' : Collinear A D E) : wedge D B E = (divratio A B C) * wedge D C E := by
+  rw[← wedge312 D B E, wedge_eq_wedge_add_wedge_of_collinear E A D B (Collinear.perm₃₁₂ colin'), ← wedge312 D C E, wedge_eq_wedge_add_wedge_of_collinear E A D C (Collinear.perm₃₁₂ colin'), wedge213 A B D, wedge312 A B E, wedge213 A C D, wedge312 A C E,wedge_eq_divratio_mul_wedge_of_collinear A B C D colin,wedge_eq_divratio_mul_wedge_of_collinear A B C E colin]
   ring
 
-theorem ratio_eq_wedge_div_wedge_of_collinear_collinear_notcoliear (A B C D E : P) (colin : collinear A B C) [cnea : PtNe C A] (colin' : collinear A D E) (ncolin : ¬ collinear D C E) : divratio A B C = (wedge D B E) / (wedge D C E) := by
+theorem ratio_eq_wedge_div_wedge_of_collinear_collinear_notcoliear (A B C D E : P) (colin : Collinear A B C) [cnea : PtNe C A] (colin' : Collinear A D E) (ncolin : ¬ Collinear D C E) : divratio A B C = (wedge D B E) / (wedge D C E) := by
   rw [wedge_eq_divratio_mul_wedge_of_collinear_collinear A B C D E colin colin']
   have h0 : ¬ wedge D C E = 0 := by
     rw[(collinear_iff_wedge_eq_zero D C E).symm]
diff --git a/EuclideanGeometry/Foundation/Construction/Polygon/Parallelogram.lean b/EuclideanGeometry/Foundation/Construction/Polygon/Parallelogram.lean
index 424b374b..53436c17 100644
--- a/EuclideanGeometry/Foundation/Construction/Polygon/Parallelogram.lean
+++ b/EuclideanGeometry/Foundation/Construction/Polygon/Parallelogram.lean
@@ -34,18 +34,50 @@ In this section we define two types of parallelogram. 'parallel_nd' deals with t
 
 @[pp_dot]
 structure Quadrilateral_nd.Parallelogram_non_triv {P : Type _} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) : Prop where
-  not_collinear₁₂₃: ( ¬ collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃)
-  not_collinear₂₃₄: ( ¬ collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄)
-  not_collinear₃₄₁: ( ¬ collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁)
-  not_collinear₄₁₂: ( ¬ collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂)
+  not_collinear₁₂₃: ( ¬ Collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃)
+  not_collinear₂₃₄: ( ¬ Collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄)
+  not_collinear₃₄₁: ( ¬ Collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁)
+  not_collinear₄₁₂: ( ¬ Collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂)
 
 /-- A quadrilateral is called parallelogram if VEC qdr.point₁ qdr.point₂ = VEC qdr.point₄ qdr.point₃.-/
 @[pp_dot]
 def Quadrilateral.IsParallelogram {P : Type _} [EuclideanPlane P] (qdr : Quadrilateral P) : Prop := VEC qdr.point₁ qdr.point₂ = VEC qdr.point₄ qdr.point₃
 
+lemma Quadrilateral.isParallelogram_def₁₂ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} :
+    qdr.IsParallelogram ↔ VEC qdr.point₁ qdr.point₂ = VEC qdr.point₄ qdr.point₃ :=
+  .rfl
+
+alias ⟨Quadrilateral.IsParallelogram.elim₁₂, _⟩ := Quadrilateral.isParallelogram_def₁₂
+
+lemma Quadrilateral.isParallelogram_def₂₁ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} :
+    qdr.IsParallelogram ↔ VEC qdr.point₂ qdr.point₁ = VEC qdr.point₃ qdr.point₄ := by
+  rw [← neg_inj, neg_vec, neg_vec, qdr.isParallelogram_def₁₂]
+
+alias ⟨Quadrilateral.IsParallelogram.elim₂₁, _⟩ := Quadrilateral.isParallelogram_def₂₁
+
+lemma Quadrilateral.isParallelogram_def₃₂ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} :
+    qdr.IsParallelogram ↔ VEC qdr.point₃ qdr.point₂ = VEC qdr.point₄ qdr.point₁ := by
+  rw [isParallelogram_def₁₂]
+  dsimp [(VEC · ·)]
+  rw [← sub_eq_zero, ← sub_eq_zero (a := _ -ᵥ _), vsub_sub_vsub_comm]
+
+alias ⟨Quadrilateral.IsParallelogram.elim₃₂, _⟩ := Quadrilateral.isParallelogram_def₃₂
+
+lemma Quadrilateral.isParallelogram_def₂₃ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} :
+    qdr.IsParallelogram ↔ VEC qdr.point₂ qdr.point₃ = VEC qdr.point₁ qdr.point₄ := by
+  rw [← neg_inj, neg_vec, neg_vec, qdr.isParallelogram_def₃₂]
+
+alias ⟨Quadrilateral.IsParallelogram.elim₂₃, _⟩ := Quadrilateral.isParallelogram_def₂₃
+
+lemma Quadrilateral.isParallelogram_def₁₄ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} :
+    qdr.IsParallelogram ↔ VEC qdr.point₁ qdr.point₄ = VEC qdr.point₂ qdr.point₃ := by
+  rw [qdr.isParallelogram_def₂₃, eq_comm]
+
+alias ⟨Quadrilateral.IsParallelogram.elim₁₄, _⟩ := Quadrilateral.isParallelogram_def₁₄
+
 /-- A quadrilateral satisfies IsParallelogram_para if two sets of opposite sides are parallel respectively. -/
 @[pp_dot]
-def Quadrilateral_nd.IsParallelogram_para {P : Type _} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) : Prop := ( qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄) ∧ (qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃)
+def Quadrilateral_nd.IsParallelogram_para {P : Type _} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) : Prop := ( qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄) ∧ (qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃)
 
 /-- A quadrilateral satisfies IsParallelogram_nd if it satisfies both IsParallelogram_para and Parallelogram_non_triv. It is now what we commonly call parallelogram.-/
 @[pp_dot]
@@ -74,18 +106,18 @@ structure Parallelogram_nd (P : Type _) [EuclideanPlane P] extends Quadrilateral
   is_parallelogram_non_triv : toQuadrilateral_nd IsParallelogram_non_triv
 
 /-- We would also like to introduce a shortcut in proving statements concerning parallelograms - that is permute, the same technique used in Quadrilateral.lean. If qdr IsParallelogram, then its permute also IsParallelogram. -/
-theorem qdr_is_parallelogram_permute_iff (P : Type _) [EuclideanPlane P] (qdr : Quadrilateral P) :(qdr.IsParallelogram) ↔ ((qdr.permute).IsParallelogram) := by
+theorem qdr_is_parallelogram_permute_iff {P : Type*} [EuclideanPlane P] (qdr : Quadrilateral P) :(qdr.IsParallelogram) ↔ ((qdr.permute).IsParallelogram) := by
   sorry
 
 /-- If qdr IsParallelogram_nd, then its permute also IsParallelogram_nd. -/
-theorem qdr_is_parallelogram_nd_permute_iff (P : Type _) [EuclideanPlane P] (qdr : Quadrilateral P) :(qdr.IsParallelogram_nd) ↔ ((qdr.permute).IsParallelogram_nd) := by
+theorem qdr_is_parallelogram_nd_permute_iff {P : Type*} [EuclideanPlane P] (qdr : Quadrilateral P) :(qdr.IsParallelogram_nd) ↔ ((qdr.permute).IsParallelogram_nd) := by
   sorry
 
 /-- If qdr_nd IsParallelogram, then its permute also IsParallelogram. -/
-theorem qdr_nd_is_parallelogram_permute_iff (P : Type _) [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) :(qdr_nd.IsParallelogram) ↔ ((qdr_nd.permute).IsParallelogram) := by sorry
+theorem qdr_nd_is_parallelogram_permute_iff {P : Type*} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) :(qdr_nd.IsParallelogram) ↔ ((qdr_nd.permute).IsParallelogram) := by sorry
 
 /-- If qdr_nd IsParallelogram_nd, then its permute also IsParallelogram. -/
-theorem qdr_nd_is_parallelogram_nd_permute_iff (P : Type _) [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) :(qdr_nd.IsParallelogram_nd) ↔ ((qdr_nd.permute).IsParallelogram_nd) := by
+theorem qdr_nd_is_parallelogram_nd_permute_iff {P : Type*} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) :(qdr_nd.IsParallelogram_nd) ↔ ((qdr_nd.permute).IsParallelogram_nd) := by
   constructor
   intro h
   unfold Quadrilateral_nd.IsParallelogram_nd
@@ -94,20 +126,20 @@ theorem qdr_nd_is_parallelogram_nd_permute_iff (P : Type _) [EuclideanPlane P] (
   unfold Quadrilateral_nd.IsParallelogram_para
   unfold Quadrilateral_nd.IsParallelogram_para at a
   rcases a with ⟨k,q⟩
-  unfold parallel
-  unfold parallel at k q
-  have K₁: qdr_nd.permute.edge_nd₁₂.toProj = qdr_nd.edge_nd₂₃.toProj := by rfl
-  have K₂: qdr_nd.permute.edge_nd₃₄.toProj = qdr_nd.edge_nd₁₄.reverse.toProj := by rfl
-  have K₃: qdr_nd.edge_nd₁₄.reverse.toProj = qdr_nd.edge_nd₁₄.toProj := by apply SegND.toProj_of_rev_eq_toProj
+  unfold Parallel
+  unfold Parallel at k q
+  have K₁: qdr_nd.permute.edgeND₁₂.toProj = qdr_nd.edgeND₂₃.toProj := by rfl
+  have K₂: qdr_nd.permute.edgeND₃₄.toProj = qdr_nd.edgeND₄₁.toProj := by rfl
+  have K₃: qdr_nd.edgeND₄₁.toProj = qdr_nd.edgeND₄₁.reverse.toProj := by apply SegND.toProj_of_rev_eq_toProj.symm
   rw [K₃] at K₂
-  have K₄: qdr_nd.permute.edge_nd₁₂.toProj = qdr_nd.permute.edge_nd₃₄.toProj := by
+  have K₄: qdr_nd.permute.edgeND₁₂.toProj = qdr_nd.permute.edgeND₃₄.toProj := by
     rw [K₁, K₂]
     exact q.symm
-  have K₅: qdr_nd.permute.edge_nd₁₄.toProj = qdr_nd.edge_nd₁₂.reverse.toProj := by rfl
-  have K₆: qdr_nd.edge_nd₁₂.reverse.toProj = qdr_nd.edge_nd₁₂.toProj := by apply SegND.toProj_of_rev_eq_toProj
+  have K₅: qdr_nd.permute.edgeND₄₁.reverse.toProj = qdr_nd.edgeND₁₂.reverse.toProj := by rfl
+  have K₆: qdr_nd.edgeND₁₂.reverse.toProj = qdr_nd.edgeND₁₂.toProj := by apply SegND.toProj_of_rev_eq_toProj
   rw [K₆] at K₅
-  have K₇: qdr_nd.permute.edge_nd₂₃.toProj = qdr_nd.edge_nd₃₄.toProj := by rfl
-  have K₈: qdr_nd.permute.edge_nd₁₄.toProj = qdr_nd.permute.edge_nd₂₃.toProj := by
+  have K₇: qdr_nd.permute.edgeND₂₃.toProj = qdr_nd.edgeND₃₄.toProj := by rfl
+  have K₈: qdr_nd.permute.edgeND₄₁.reverse.toProj = qdr_nd.permute.edgeND₂₃.toProj := by
     rw [K₅, K₇]
     exact k
   constructor
@@ -127,15 +159,15 @@ theorem qdr_nd_is_parallelogram_nd_permute_iff (P : Type _) [EuclideanPlane P] (
   unfold Quadrilateral_nd.IsParallelogram_para
   unfold Quadrilateral_nd.IsParallelogram_para at a
   rcases a with ⟨k,q⟩
-  unfold parallel
-  unfold parallel at k q
-  have K₂: qdr_nd.permute.edge_nd₃₄.toProj = qdr_nd.edge_nd₁₄.reverse.toProj := by rfl
-  have K₃: qdr_nd.edge_nd₁₄.reverse.toProj = qdr_nd.edge_nd₁₄.toProj := by apply SegND.toProj_of_rev_eq_toProj
+  unfold Parallel
+  unfold Parallel at k q
+  have K₂: qdr_nd.permute.edgeND₃₄.toProj = qdr_nd.edgeND₄₁.toProj := by rfl
+  have K₃: qdr_nd.edgeND₄₁.toProj = qdr_nd.edgeND₄₁.reverse.toProj := by apply SegND.toProj_of_rev_eq_toProj.symm
   rw [K₃] at K₂
-  have K₅: qdr_nd.permute.edge_nd₁₄.toProj = qdr_nd.edge_nd₁₂.reverse.toProj := by rfl
-  have K₆: qdr_nd.edge_nd₁₂.reverse.toProj = qdr_nd.edge_nd₁₂.toProj := by apply SegND.toProj_of_rev_eq_toProj
+  have K₅: qdr_nd.permute.edgeND₄₁.reverse.toProj = qdr_nd.edgeND₁₂.reverse.toProj := by rfl
+  have K₆: qdr_nd.edgeND₁₂.reverse.toProj = qdr_nd.edgeND₁₂.toProj := by apply SegND.toProj_of_rev_eq_toProj
   rw [K₆] at K₅
-  have K₇: qdr_nd.permute.edge_nd₂₃.toProj = qdr_nd.edge_nd₃₄.toProj := by rfl
+  have K₇: qdr_nd.permute.edgeND₂₃.toProj = qdr_nd.edgeND₃₄.toProj := by rfl
   constructor
   constructor
   exact (Eq.congr (id K₅.symm) K₇).mpr q
@@ -148,178 +180,45 @@ theorem qdr_nd_is_parallelogram_nd_permute_iff (P : Type _) [EuclideanPlane P] (
   exact z
 
 /-- If prg IsParallelogram_nd, then its permute also IsParallelogram_nd. -/
-theorem prg_is_parallelogram_nd_permute_iff (P : Type _) [EuclideanPlane P] (prg : Parallelogram P) :(prg.IsParallelogram_nd) ↔ ((prg.permute).IsParallelogram_nd) := by
+theorem prg_is_parallelogram_nd_permute_iff {P : Type*} [EuclideanPlane P] (prg : Parallelogram P) :(prg.IsParallelogram_nd) ↔ ((prg.permute).IsParallelogram_nd) := by
   sorry
 
 /-- If qdr_nd is non-degenerate and is a parallelogram, and its 1st, 2nd and 3rd points are not collinear, then qdr_nd is a parallelogram_nd.-/
-theorem Parallelogram_not_collinear₁₂₃ (P : Type _) [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃) : qdr_nd IsParallelogram_nd := by
-
-   have hba : qdr_nd.point₂ ≠ qdr_nd.point₁ := by
-     unfold collinear at h
-     exact (ne_of_not_collinear h).2.2
-   have hbc : qdr_nd.point₂ ≠ qdr_nd.point₃ :=by
-     exact (ne_of_not_collinear h).1.symm
-   have hca: qdr_nd.point₃ ≠ qdr_nd.point₁ :=by
-     exact (ne_of_not_collinear h).2.1.symm
-   have had : qdr_nd.point₁ ≠ qdr_nd.point₄ := by
-     by_contra k
-     unfold Quadrilateral.IsParallelogram at para
-     have o : VEC qdr_nd.point₁ qdr_nd.point₂ = VEC qdr_nd.point₁ qdr_nd.point₃ := by
-       simp only [para, k.symm]
-     rw [EuclidGeom.Vec.mkPtPt] at o
-     rw [EuclidGeom.Vec.mkPtPt] at o
-     have oo : qdr_nd.point₂= qdr_nd.point₃ := by
-       exact vsub_left_cancel o
-     exact hbc oo
-
-   have hbd : qdr_nd.point₂ ≠ qdr_nd.point₄ := by
-     by_contra k₁
-     unfold Quadrilateral.IsParallelogram at para
-     have o: VEC qdr_nd.point₁ qdr_nd.point₂ = VEC qdr_nd.point₂ qdr_nd.point₃ := by
-       simp[para,k₁.symm]
-     have oo: collinear qdr_nd.point₂ qdr_nd.point₁ qdr_nd.point₃ := by
-       have ooo:∃ t : ℝ,VEC qdr_nd.point₂ qdr_nd.point₃ = t • VEC qdr_nd.point₂ qdr_nd.point₁ := by
-         use -1
-         simp only [neg_smul, one_smul, neg_vec]
-         rw[o]
-       exact collinear_of_vec_eq_smul_vec' ooo
-     simp only [flip_collinear_fst_snd oo, not_true_eq_false] at h
-   have hcd : qdr_nd.point₃ ≠ qdr_nd.point₄ := by
-     by_contra k₂
-     have k₂₂ : VEC qdr_nd.point₄ qdr_nd.point₃=0 := by
-       simp only [k₂, vec_same_eq_zero]
-     unfold Quadrilateral.IsParallelogram at para
-     have k₂₃: qdr_nd.point₁=qdr_nd.point₂ :=by
-       rw [k₂₂] at para
-       simp only [para, (eq_iff_vec_eq_zero qdr_nd.point₁ qdr_nd.point₂).mpr, vec_same_eq_zero]
-     simp only [k₂₃.symm, ne_eq, not_true_eq_false] at hba
-   have t : ¬ collinear qdr_nd.point₂ qdr_nd.point₁ qdr_nd.point₃ := by
-     by_contra k
-     simp only [flip_collinear_fst_snd k, not_true_eq_false] at h
-   have x : ¬ collinear_of_nd (A := qdr_nd.point₂) (B := qdr_nd.point₁) (C := qdr_nd.point₃) := by
-     unfold collinear at t
-     simp only [hca, hbc, hba.symm, or_self, dite_false] at t
-     simp only [t, not_false_eq_true]
-   have l₁ : qdr_nd.edge_nd₁₂.toProj=VecND.toProj ⟨VEC qdr_nd.point₁ qdr_nd.point₂, _⟩ := by rfl
-   have l₁' : qdr_nd.edge_nd₁₂.toProj=VecND.toProj ⟨VEC qdr_nd.point₂ qdr_nd.point₁, (ne_iff_vec_ne_zero _ _).mp hba.symm⟩ := by
-     have y₁:VecND.toProj ⟨VEC qdr_nd.point₂ qdr_nd.point₁, (ne_iff_vec_ne_zero _ _).mp hba.symm⟩=VecND.toProj ⟨VEC qdr_nd.point₁ qdr_nd.point₂, (ne_iff_vec_ne_zero _ _).mp hba⟩ := by
-       have z₁: VEC qdr_nd.point₂ qdr_nd.point₁ =- VEC qdr_nd.point₁ qdr_nd.point₂ := by
-         simp only [neg_vec]
-       simp only [ne_eq, z₁, VecND.mk_neg', VecND.neg_toProj]
-     simp only [l₁, ne_eq, y₁]
-   have l₂ : qdr_nd.edge_nd₂₃.toProj=VecND.toProj ⟨VEC qdr_nd.point₂ qdr_nd.point₃, _⟩ := by rfl
-   have l₂' : qdr_nd.edge_nd₂₃.toProj=VecND.toProj ⟨VEC qdr_nd.point₃ qdr_nd.point₂, (ne_iff_vec_ne_zero _ _).mp hbc⟩ := by
-     have y₂:VecND.toProj ⟨VEC qdr_nd.point₃ qdr_nd.point₂, (ne_iff_vec_ne_zero _ _).mp hbc⟩=VecND.toProj ⟨VEC qdr_nd.point₂ qdr_nd.point₃, (ne_iff_vec_ne_zero _ _).mp hbc.symm⟩ := by
-       have z₂: VEC qdr_nd.point₃ qdr_nd.point₂ =- VEC qdr_nd.point₂ qdr_nd.point₃ := by
-         simp only[neg_vec]
-       simp only [ne_eq, z₂, VecND.mk_neg', VecND.neg_toProj]
-     simp only [l₂, ne_eq, y₂]
-   have l₃ : qdr_nd.edge_nd₃₄.toProj=VecND.toProj ⟨VEC qdr_nd.point₃ qdr_nd.point₄, _⟩ := by rfl
-   have l₃' : qdr_nd.edge_nd₃₄.toProj=VecND.toProj ⟨VEC qdr_nd.point₄ qdr_nd.point₃, (ne_iff_vec_ne_zero _ _).mp hcd⟩ := by
-     have y₃:VecND.toProj ⟨VEC qdr_nd.point₄ qdr_nd.point₃, (ne_iff_vec_ne_zero _ _).mp hcd⟩=VecND.toProj ⟨VEC qdr_nd.point₃ qdr_nd.point₄, (ne_iff_vec_ne_zero _ _).mp hcd.symm⟩ := by
-       have z₃: VEC qdr_nd.point₄ qdr_nd.point₃ =- VEC qdr_nd.point₃ qdr_nd.point₄ := by
-         simp only [neg_vec]
-       simp only [ne_eq, z₃, VecND.mk_neg', VecND.neg_toProj]
-     simp only [l₃, ne_eq, y₃]
-   have l₄ : qdr_nd.edge_nd₁₄.toProj=VecND.toProj ⟨VEC qdr_nd.point₁ qdr_nd.point₄, _⟩ := by rfl
-   have l₄' : qdr_nd.edge_nd₁₄.toProj=VecND.toProj ⟨VEC qdr_nd.point₄ qdr_nd.point₁, (ne_iff_vec_ne_zero _ _).mp had⟩ := by
-     have y₄:VecND.toProj ⟨VEC qdr_nd.point₄ qdr_nd.point₁, (ne_iff_vec_ne_zero _ _).mp had⟩=VecND.toProj ⟨VEC qdr_nd.point₁ qdr_nd.point₄, (ne_iff_vec_ne_zero _ _).mp had.symm⟩ := by
-       have z₄: VEC qdr_nd.point₄ qdr_nd.point₁ =- VEC qdr_nd.point₁ qdr_nd.point₄ := by
-         simp [neg_vec]
-       simp only [ne_eq, z₄, VecND.mk_neg', VecND.neg_toProj]
-     simp only [l₄, ne_eq, y₄]
-   have s : ¬ qdr_nd.edge_nd₁₂.toProj = qdr_nd.edge_nd₂₃.toProj := by
-     unfold collinear_of_nd at x
-     simp only [l₁', ne_eq, l₂]
-     exact x
-   have v₁ : qdr_nd.edge_nd₁₂.toProj = qdr_nd.edge_nd₃₄.toProj := by
-     unfold Quadrilateral.IsParallelogram at para
-     have v₁₁₁:VecND.toProj ⟨VEC qdr_nd.point₁ qdr_nd.point₂, (ne_iff_vec_ne_zero _ _).mp hba⟩ = VecND.toProj ⟨VEC qdr_nd.point₄ qdr_nd.point₃, (ne_iff_vec_ne_zero _ _).mp hcd⟩ := by
-       simp only [ne_eq, para]
-     simp only [l₁, ne_eq, v₁₁₁, l₃']
-   have v₁₁ : toProj qdr_nd.edge_nd₁₂ = toProj qdr_nd.edge_nd₃₄ := by exact v₁
-   have v₂ : qdr_nd.edge_nd₂₃.toProj = qdr_nd.edge_nd₁₄.toProj := by
-     unfold Quadrilateral.IsParallelogram at para
-     have v₂₂: VEC qdr_nd.point₁ qdr_nd.point₄ = VEC qdr_nd.point₂ qdr_nd.point₃ := by
-       rw [← vec_add_vec qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₄]
-       rw [← vec_add_vec qdr_nd.point₂ qdr_nd.point₄ qdr_nd.point₃]
-       rw [para]
-       exact add_comm (VEC qdr_nd.point₄ qdr_nd.point₃) (VEC qdr_nd.point₂ qdr_nd.point₄)
-     simp only [l₂, ne_eq, v₂₂, l₄]
-   have v₂₁ : toProj qdr_nd.edge_nd₂₃ = toProj qdr_nd.edge_nd₁₄ := by exact v₂
-   have s₁ : ¬ qdr_nd.edge_nd₁₂.toProj = qdr_nd.edge_nd₂₃.toProj := by
-     by_contra k₃
-     have k₃₃: collinear_of_nd (A := qdr_nd.point₂) (B := qdr_nd.point₁) (C := qdr_nd.point₃) := by
-        rw[l₁',l₂] at k₃
-        exact k₃
-     have k₃₄: collinear qdr_nd.point₂ qdr_nd.point₁ qdr_nd.point₃ := by
-       unfold collinear
-       simp only [hca, hbc, hba.symm, or_self, k₃₃, dite_eq_ite, ite_self]
-     simp only [k₃₄, not_true_eq_false] at t
-   have s₂ : ¬ qdr_nd.edge_nd₂₃.toProj = qdr_nd.edge_nd₃₄.toProj := by
-     rw [v₁.symm]
-     by_contra k
-     exact s k.symm
-   have s₃ : ¬ qdr_nd.edge_nd₃₄.toProj = qdr_nd.edge_nd₁₄.toProj := by
-     rw [v₂.symm]
-     by_contra k
-     exact s₂ k.symm
-   have s₄ : ¬ qdr_nd.edge_nd₁₄.toProj = qdr_nd.edge_nd₁₂.toProj := by
-     rw[v₁]
-     by_contra k
-     exact s₃ k.symm
-   constructor
-   constructor
-   unfold parallel
-   simp only [v₁₁]
-   unfold parallel
-   simp only [v₂₁]
-   constructor
-   simp only [h, not_false_eq_true]
-   by_contra m₃
-   have m₂ :  ¬ collinear qdr_nd.point₃ qdr_nd.point₂ qdr_nd.point₄ := by
-     by_contra k₁
-     unfold collinear at k₁
-     simp [hbd.symm,hcd,hbc] at k₁
-     unfold collinear_of_nd at k₁
-     have p₄:qdr_nd.edge_nd₂₃.toProj = qdr_nd.edge_nd₃₄.toProj  := by
-       rw[l₂',l₃]
-       exact k₁
-     simp[p₄] at s₂
-   simp [flip_collinear_fst_snd m₃] at m₂
-   by_contra m₅
-   have m₄ :  ¬ collinear qdr_nd.point₄ qdr_nd.point₃ qdr_nd.point₁ := by
-     by_contra k₂
-     unfold collinear at k₂
-     simp [hca.symm,hcd,had.symm] at k₂
-     unfold collinear_of_nd at k₂
-     have p₅:qdr_nd.edge_nd₃₄.toProj = qdr_nd.edge_nd₁₄.toProj := by
-       rw[l₄',l₃']
-       exact k₂
-     simp[p₅] at s₃
-   simp [flip_collinear_fst_snd m₅] at m₄
-   by_contra m₇
-   have m₆ :  ¬ collinear qdr_nd.point₁ qdr_nd.point₄ qdr_nd.point₂ := by
-     by_contra k₃
-     unfold collinear at k₃
-     simp [hbd,hba.symm,had.symm] at k₃
-     unfold collinear_of_nd at k₃
-     have p₆:qdr_nd.edge_nd₁₄.toProj = qdr_nd.edge_nd₁₂.toProj := by
-       rw[l₄,l₁]
-       exact k₃
-     simp [p₆] at s₄
-   simp [flip_collinear_fst_snd m₇] at m₆
+theorem Parallelogram_not_collinear₁₂₃ {P : Type*} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ Collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃) : qdr_nd IsParallelogram_nd := by
+  constructor
+  · constructor
+    · rw [parallel_iff_wedge_eq_wedge_left]
+      dsimp
+      rw [← wedge_smul_vec_self_vadd _ _ _ (-1), para]
+      simp
+    · rw [parallel_iff_wedge_eq_wedge_left]
+      dsimp
+      rw [← wedge_smul_vec_self_vadd _ _ _ 1, para.elim₁₄]
+      simp [wedge132 _ _ qdr_nd.point₄]
+  · have h₂₃₄ : ¬Collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄
+    · rwa [collinear231, collinear_iff_collinear] at h
+      use 1
+      rw [para.elim₁₄, one_smul]
+    have h₃₄₁ : ¬Collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁
+    · rwa [collinear231, collinear_iff_collinear] at h₂₃₄
+      use 1
+      rw [para.elim₂₁, one_smul]
+    have h₄₁₂ : ¬Collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂
+    · rwa [collinear231, collinear_iff_collinear] at h₃₄₁
+      use 1
+      rw [para.elim₃₂, one_smul]
+    constructor <;> assumption
 
 /-- If qdr_nd is non-degenerate and is a parallelogram, and its 2nd, 3rd and 4th points are not collinear, then qdr_nd is a parallelogram_nd.-/
-theorem Parallelogram_not_collinear₂₃₄ (P : Type _) [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄) : qdr_nd IsParallelogram_nd := by
+theorem Parallelogram_not_collinear₂₃₄ {P : Type*} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ Collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄) : qdr_nd IsParallelogram_nd := by
   sorry
 
 /-- If qdr_nd is non-degenerate and is a parallelogram, and its 3rd, 4th and 1st points are not collinear, then qdr_nd is a parallelogram_nd.-/
-theorem Parallelogram_not_collinear₃₄₁ (P : Type _) [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁) : qdr_nd IsParallelogram_nd := by
+theorem Parallelogram_not_collinear₃₄₁ {P : Type*} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ Collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁) : qdr_nd IsParallelogram_nd := by
   sorry
 
 /-- If qdr_nd is non-degenerate and is a parallelogram, and its 4th, 1st and 2nd points are not collinear, then qdr_nd is a parallelogram_nd.-/
-theorem Parallelogram_not_collinear₄₁₂ (P : Type _) [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂) : qdr_nd IsParallelogram_nd := by
+theorem Parallelogram_not_collinear₄₁₂ {P : Type*} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P) (para: qdr_nd.1 IsParallelogram) (h: ¬ Collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂) : qdr_nd IsParallelogram_nd := by
   sorry
 
 /-- Make a parallelogram with 4 points on a plane.-/
@@ -328,7 +227,7 @@ def Parallelogram.mk_pt_pt_pt_pt {P : Type _} [EuclideanPlane P] (A B C D : P) (
   is_parallelogram := h
 
 /-- Make a parallelogram_nd with 4 points on a plane, and using condition non_collinear₁₂₃.-/
-def Parallelogram_nd.mk_pt_pt_pt_pt₄ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₁₂₃: ¬ collinear A B C) : Parallelogram_nd P where
+def Parallelogram_nd.mk_pt_pt_pt_pt₄ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₁₂₃: ¬ Collinear A B C) : Parallelogram_nd P where
   toQuadrilateral := (QDR A B C D)
   is_parallelogram := h'
   nd := h
@@ -336,7 +235,7 @@ def Parallelogram_nd.mk_pt_pt_pt_pt₄ {P : Type _} [EuclideanPlane P] (A B C D
   is_parallelogram_non_triv := sorry
 
 /-- Make a parallelogram_nd with 4 points on a plane, and using condition non_collinear₂₃₄.-/
-def Parallelogram_nd.mk_pt_pt_pt_pt₁ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₂₃₄: ¬ collinear B C D) : Parallelogram_nd P where
+def Parallelogram_nd.mk_pt_pt_pt_pt₁ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₂₃₄: ¬ Collinear B C D) : Parallelogram_nd P where
   toQuadrilateral := (QDR A B C D)
   is_parallelogram := h'
   nd := h
@@ -344,7 +243,7 @@ def Parallelogram_nd.mk_pt_pt_pt_pt₁ {P : Type _} [EuclideanPlane P] (A B C D
   is_parallelogram_non_triv := sorry
 
 /-- Make a parallelogram_nd with 4 points on a plane, and using condition non_collinear₃₄₁.-/
-def Parallelogram_nd.mk_pt_pt_pt_pt₂ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₃₄₁: ¬ collinear C D A) : Parallelogram_nd P where
+def Parallelogram_nd.mk_pt_pt_pt_pt₂ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₃₄₁: ¬ Collinear C D A) : Parallelogram_nd P where
   toQuadrilateral := (QDR A B C D)
   is_parallelogram := h'
   nd := h
@@ -352,7 +251,7 @@ def Parallelogram_nd.mk_pt_pt_pt_pt₂ {P : Type _} [EuclideanPlane P] (A B C D
   is_parallelogram_non_triv := sorry
 
 /-- Make a parallelogram_nd with 4 points on a plane, and using condition non_collinear₄₁₂.-/
-def Parallelogram_nd.mk_pt_pt_pt_pt₃ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₄₁₂: ¬ collinear D A B) : Parallelogram_nd P where
+def Parallelogram_nd.mk_pt_pt_pt_pt₃ {P : Type _} [EuclideanPlane P] (A B C D : P) (h : (QDR A B C D).IsND) (h': (QDR A B C D) IsParallelogram) (non_collinear₄₁₂: ¬ Collinear D A B) : Parallelogram_nd P where
   toQuadrilateral := (QDR A B C D)
   is_parallelogram := h'
   nd := h
@@ -380,7 +279,7 @@ def mk_parallelogram {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h
   is_parallelogram := h
 
 /-- Make parallelogram_nd with a quadrilateral, using condition non_collinear₁₂₃.-/
-def mk_parallelogram_nd₄ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₁₂₃: ¬ collinear qdr.1 qdr.2 qdr.3) : Parallelogram_nd P where
+def mk_parallelogram_nd₄ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₁₂₃: ¬ Collinear qdr.1 qdr.2 qdr.3) : Parallelogram_nd P where
   toQuadrilateral := qdr
   is_parallelogram := h
   nd := h'
@@ -388,7 +287,7 @@ def mk_parallelogram_nd₄ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral
   is_parallelogram_non_triv := sorry
 
 /-- Make parallelogram_nd with a quadrilateral, using condition non_collinear₂₃₄.-/
-def mk_parallelogram_nd₁ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₂₃₄: ¬ collinear qdr.2 qdr.3 qdr.4) : Parallelogram_nd P where
+def mk_parallelogram_nd₁ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₂₃₄: ¬ Collinear qdr.2 qdr.3 qdr.4) : Parallelogram_nd P where
   toQuadrilateral := qdr
   is_parallelogram := h
   nd := h'
@@ -396,7 +295,7 @@ def mk_parallelogram_nd₁ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral
   is_parallelogram_non_triv := sorry
 
 /-- Make parallelogram_nd with a quadrilateral, using condition non_collinear₃₁₄.-/
-def mk_parallelogram_nd₂ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₃₄₁: ¬ collinear qdr.3 qdr.4 qdr.1) : Parallelogram_nd P where
+def mk_parallelogram_nd₂ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₃₄₁: ¬ Collinear qdr.3 qdr.4 qdr.1) : Parallelogram_nd P where
   toQuadrilateral := qdr
   is_parallelogram := h
   nd := h'
@@ -404,7 +303,7 @@ def mk_parallelogram_nd₂ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral
   is_parallelogram_non_triv := sorry
 
 /-- Make parallelogram_nd with a quadrilateral, using condition non_collinear₄₁₂.-/
-def mk_parallelogram_nd₃ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₄₁₂: ¬ collinear qdr.4 qdr.1 qdr.2) : Parallelogram_nd P where
+def mk_parallelogram_nd₃ {P : Type _} [EuclideanPlane P] {qdr : Quadrilateral P} (h : qdr IsParallelogram) (h': qdr.IsND) (non_collinear₄₁₂: ¬ Collinear qdr.4 qdr.1 qdr.2) : Parallelogram_nd P where
   toQuadrilateral := qdr
   is_parallelogram := h
   nd := h'
@@ -442,8 +341,8 @@ variable {P : Type _} [EuclideanPlane P]
 variable {A B C D : P} (nd : (QDR A B C D).IsND) (cvx : (QDR A B C D).IsConvex)
 variable (qdr : Quadrilateral P) (qdr_nd : Quadrilateral_nd P) (qdr_cvx : Quadrilateral_cvx P)
 
-/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄, qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃, and qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄) (h₂ : qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃) (notcollinear : ¬ collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃) : qdr_nd.IsParallelogram_nd := by
+/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄, qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃, and qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃ is not collinear, then qdr_nd is a Parallelogram_nd. -/
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄) (h₂ : qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃) (notcollinear : ¬ Collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃) : qdr_nd.IsParallelogram_nd := by
   have para_vec (Vec1 Vec2 : VecND) (para : Vec1 ∥ Vec2) : ∃ c : ℝ , (Vec1 : Vec) = c • (Vec2 : Vec) := by
         apply VecND.toProj_eq_toProj_iff.mp
         exact para
@@ -455,9 +354,9 @@ theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edge
     rw [collinear_iff_eq_smul_vec_of_ne] at iscollinear
     rcases iscollinear with ⟨r,eq⟩
     apply notcollinear
-    have para : qdr_nd.edge_nd₁₄.toVecND ∥ qdr_nd.edge_nd₂₃.toVecND := by sorry
+    have para : qdr_nd.edgeND₄₁.reverse.toVecND ∥ qdr_nd.edgeND₂₃.toVecND := by sorry
     rw [collinear_iff_eq_smul_vec_of_ne]
-    rcases (para_vec qdr_nd.edge_nd₁₄.toVecND qdr_nd.edge_nd₂₃.toVecND para) with ⟨c,eq'⟩
+    rcases (para_vec qdr_nd.edgeND₄₁.reverse.toVecND qdr_nd.edgeND₂₃.toVecND para) with ⟨c,eq'⟩
     have eq'' : (VEC qdr_nd.point₁ qdr_nd.point₄) = c • (VEC qdr_nd.point₂ qdr_nd.point₃) := eq'
     have l₁ : (VEC qdr_nd.point₁ qdr_nd.point₂) = (c - r) • (VEC qdr_nd.point₂ qdr_nd.point₃) := by
       rw [(vec_sub_vec' qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂).symm]
@@ -470,7 +369,7 @@ theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edge
       by_contra is_zero
       rw [is_zero] at l₁
       have not_nd₁₂ : qdr_nd.point₂ = qdr_nd.point₁ := by
-        apply (eq_iff_vec_eq_zero qdr_nd.point₁ qdr_nd.point₂).mpr
+        apply eq_iff_vec_eq_zero.mpr
         rw [l₁]
         exact zero_smul ℝ (VEC qdr_nd.point₂ qdr_nd.point₃)
       exact qdr_nd.nd₁₂.out not_nd₁₂
@@ -484,21 +383,21 @@ theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edge
     rw [collinear_iff_eq_smul_vec_of_ne]
     rw [collinear_iff_eq_smul_vec_of_ne] at iscollinear
     rcases iscollinear with ⟨r,eq⟩
-    have para : qdr_nd.edge_nd₃₄.toVecND ∥ qdr_nd.edge_nd₁₂.toVecND := by sorry
-    rcases (para_vec qdr_nd.edge_nd₃₄.toVecND qdr_nd.edge_nd₁₂.toVecND para) with ⟨c,eq'⟩
+    have para : qdr_nd.edgeND₃₄.toVecND ∥ qdr_nd.edgeND₁₂.toVecND := by sorry
+    rcases (para_vec qdr_nd.edgeND₃₄.toVecND qdr_nd.edgeND₁₂.toVecND para) with ⟨c,eq'⟩
     have eq'' : (VEC qdr_nd.point₃ qdr_nd.point₄) = c • (VEC qdr_nd.point₁ qdr_nd.point₂) := eq'
     use - r * c
     rw [(smul_smul (- r) c (VEC qdr_nd.point₁ qdr_nd.point₂)).symm,eq''.symm,neg_smul,eq.symm]
     exact (neg_vec qdr_nd.point₃ qdr_nd.point₁).symm
   by_contra iscollinear
-  apply flip_collinear_fst_trd at iscollinear
+  apply Collinear.perm₃₂₁ at iscollinear
   rw [collinear_iff_eq_smul_vec_of_ne] at iscollinear
   rcases iscollinear with ⟨r,eq⟩
   apply notcollinear
-  have para : qdr_nd.edge_nd₃₄.toVecND ∥ qdr_nd.edge_nd₁₂.toVecND := by sorry
-  apply flip_collinear_fst_trd
+  have para : qdr_nd.edgeND₃₄.toVecND ∥ qdr_nd.edgeND₁₂.toVecND := by sorry
+  apply Collinear.perm₃₂₁
   rw [collinear_iff_eq_smul_vec_of_ne]
-  rcases (para_vec qdr_nd.edge_nd₃₄.toVecND qdr_nd.edge_nd₁₂.toVecND para) with ⟨c,eq'⟩
+  rcases (para_vec qdr_nd.edgeND₃₄.toVecND qdr_nd.edgeND₁₂.toVecND para) with ⟨c,eq'⟩
   have eq'' : (VEC qdr_nd.point₃ qdr_nd.point₄) = c • (VEC qdr_nd.point₁ qdr_nd.point₂) := eq'
   have l₁ : (VEC qdr_nd.point₃ qdr_nd.point₂) = (r + c) • (VEC qdr_nd.point₁ qdr_nd.point₂) := by
     rw [(vec_sub_vec' qdr_nd.point₄ qdr_nd.point₃ qdr_nd.point₂).symm]
@@ -512,7 +411,7 @@ theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edge
     by_contra is_zero
     rw [is_zero] at l₁
     have not_nd₂₃ : qdr_nd.point₂ = qdr_nd.point₃ := by
-      apply (eq_iff_vec_eq_zero qdr_nd.point₃ qdr_nd.point₂).mpr
+      apply eq_iff_vec_eq_zero.mpr
       rw [l₁]
       exact zero_smul ℝ (VEC qdr_nd.point₁ qdr_nd.point₂)
     exact qdr_nd.nd₂₃.out not_nd₂₃.symm
@@ -523,58 +422,58 @@ theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (h₁ : qdr_nd.edge
   exact ne_zero t
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, AB ∥ CD, AD ∥ BC, and A B C is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ collinear A B C) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ Collinear A B C) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
-/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄, qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃, and qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄ (h₁ : qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄) (h₂ : qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃) (notcollinear : ¬ collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄) : qdr_nd.IsParallelogram_nd := (qdr_nd_is_parallelogram_nd_permute_iff P qdr_nd).mpr (qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ qdr_nd.permute (SegND.para_rev_of_para h₂.symm) (SegND.para_rev_of_para h₁.symm).symm notcollinear)
+/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄, qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃, and qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄ is not collinear, then qdr_nd is a Parallelogram_nd. -/
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄ (h₁ : qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄) (h₂ : qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃) (notcollinear : ¬ Collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄) : qdr_nd.IsParallelogram_nd := (qdr_nd_is_parallelogram_nd_permute_iff qdr_nd).mpr (qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃ qdr_nd.permute (SegND.para_rev_of_para h₂.symm) (SegND.para_rev_of_para h₁.symm).symm notcollinear)
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, AB ∥ CD, AD ∥ BC, and B C D is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ collinear B C D) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ Collinear B C D) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
-/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄, qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃, and qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁ (h₁ : qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄) (h₂ : qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃) (notcollinear : ¬ collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁) : qdr_nd.IsParallelogram_nd := (qdr_nd_is_parallelogram_nd_permute_iff P qdr_nd).mpr (qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄ qdr_nd.permute (SegND.para_rev_of_para h₂.symm) (SegND.para_rev_of_para h₁.symm).symm notcollinear)
+/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄, qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃, and qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁ is not collinear, then qdr_nd is a Parallelogram_nd. -/
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁ (h₁ : qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄) (h₂ : qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃) (notcollinear : ¬ Collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁) : qdr_nd.IsParallelogram_nd := (qdr_nd_is_parallelogram_nd_permute_iff qdr_nd).mpr (qdr_nd_is_prg_nd_of_para_para_not_collinear₂₃₄ qdr_nd.permute (SegND.para_rev_of_para h₂.symm) (SegND.para_rev_of_para h₁.symm).symm notcollinear)
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, AB ∥ CD, AD ∥ BC, and C D A is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ collinear C D A) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ Collinear C D A) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
-/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄, qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃, and qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₄₁₂ (h₁ : qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄) (h₂ : qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃) (notcollinear : ¬ collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂) : qdr_nd.IsParallelogram_nd := (qdr_nd_is_parallelogram_nd_permute_iff P qdr_nd).mpr (qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁ qdr_nd.permute (SegND.para_rev_of_para h₂.symm) (SegND.para_rev_of_para h₁.symm).symm notcollinear)
+/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄, qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃, and qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂ is not collinear, then qdr_nd is a Parallelogram_nd. -/
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₄₁₂ (h₁ : qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄) (h₂ : qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃) (notcollinear : ¬ Collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂) : qdr_nd.IsParallelogram_nd := (qdr_nd_is_parallelogram_nd_permute_iff qdr_nd).mpr (qdr_nd_is_prg_nd_of_para_para_not_collinear₃₄₁ qdr_nd.permute (SegND.para_rev_of_para h₂.symm) (SegND.para_rev_of_para h₁.symm).symm notcollinear)
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, AB ∥ CD, AD ∥ BC, and D A B is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₄₁₂_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ collinear D A B) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₄₁₂ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_para_para_not_collinear₄₁₂_variant (h₁ : (SEG_nd A B (QDR_nd A B C D nd).nd₁₂.out) ∥ (SEG_nd C D (QDR_nd A B C D nd).nd₃₄.out)) (h₂ : (SEG_nd A D (QDR_nd A B C D nd).nd₁₄.out) ∥ (SEG_nd B C (QDR_nd A B C D nd).nd₂₃.out)) (notcollinear : ¬ Collinear D A B) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_para_para_not_collinear₄₁₂ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
 /-- Given Quadrilateral_nd qdr_nd, qdr_nd.angle₁.value = qdr_nd.angle₃.value, qdr_nd.angle₂.value = qdr_nd.angle₄.value, and qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₁₂₃ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃) : qdr_nd.IsParallelogram_nd := by
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₁₂₃ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ Collinear qdr_nd.point₁ qdr_nd.point₂ qdr_nd.point₃) : qdr_nd.IsParallelogram_nd := by
   apply qdr_nd_is_prg_nd_of_para_para_not_collinear₁₂₃
   sorry
   sorry
   exact notcollinear
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, ∠DAB = ∠BCD, ∠ABC = ∠CDA, and A B C is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₁₂₃_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ collinear A B C) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₁₂₃ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₁₂₃_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ Collinear A B C) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₁₂₃ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
 /-- Given Quadrilateral_nd qdr_nd, qdr_nd.angle₁.value = qdr_nd.angle₃.value, qdr_nd.angle₂.value = qdr_nd.angle₄.value, and qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₂₃₄ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄) : qdr_nd.IsParallelogram_nd := by sorry
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₂₃₄ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ Collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄) : qdr_nd.IsParallelogram_nd := by sorry
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, ∠DAB = ∠BCD, ∠ABC = ∠CDA, and B C D is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₂₃₄_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ collinear B C D) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₂₃₄ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₂₃₄_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ Collinear B C D) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₂₃₄ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
 /-- Given Quadrilateral_nd qdr_nd, qdr_nd.angle₁.value = qdr_nd.angle₃.value, qdr_nd.angle₂.value = qdr_nd.angle₄.value, and qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₃₄₁ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁) : qdr_nd.IsParallelogram_nd := by sorry
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₃₄₁ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ Collinear qdr_nd.point₃ qdr_nd.point₄ qdr_nd.point₁) : qdr_nd.IsParallelogram_nd := by sorry
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, ∠DAB = ∠BCD, ∠ABC = ∠CDA, and C D A is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₃₄₁_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ collinear C D A) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₃₄₁ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₃₄₁_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ Collinear C D A) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₃₄₁ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
 /-- Given Quadrilateral_nd qdr_nd, qdr_nd.angle₁.value = qdr_nd.angle₃.value, qdr_nd.angle₂.value = qdr_nd.angle₄.value, and qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂ is not collinear, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₄₁₂ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂) : qdr_nd.IsParallelogram_nd := by sorry
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₄₁₂ (h₁ : qdr_nd.angle₁.value = qdr_nd.angle₃.value) (h₂ : qdr_nd.angle₂.value = qdr_nd.angle₄.value) (notcollinear : ¬ Collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂) : qdr_nd.IsParallelogram_nd := by sorry
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, ∠DAB = ∠BCD, ∠ABC = ∠CDA, and D A B is not collinear, then Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₄₁₂_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ collinear D A B) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₄₁₂ (QDR_nd A B C D nd) h₁ h₂ notcollinear
+theorem qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₄₁₂_variant (h₁ : (ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value = (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value) (h₂ : (ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value = (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value) (notcollinear : ¬ Collinear D A B) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_angle_value_eq_angle_value_not_collinear₄₁₂ (QDR_nd A B C D nd) h₁ h₂ notcollinear
 
-/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edge_nd₁₂.length = qdr_nd.edge_nd₃₄.length, qdr_nd.edge_nd₁₄.length = qdr_nd.edge_nd₂₃.length, the signs of qdr_nd.angle₁ and qdr_nd.angle₃ are equal, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign (h₁ : qdr_nd.edge_nd₁₂.length = qdr_nd.edge_nd₃₄.length) (h₂ : qdr_nd.edge_nd₁₄.length = qdr_nd.edge_nd₂₃.length) (h : (qdr_nd.angle₁.value.IsPos ∧ qdr_nd.angle₃.value.IsPos) ∨ (qdr_nd.angle₁.value.IsNeg ∧ qdr_nd.angle₃.value.IsNeg)) : qdr_nd.IsParallelogram_nd := by
-  have nontriv₄₁₂ : ¬ collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂ := by sorry
-  have nontriv₂₃₄ : ¬ collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄ := by sorry
+/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edgeND₁₂.length = qdr_nd.edgeND₃₄.length, qdr_nd.edgeND₄₁.reverse.length = qdr_nd.edgeND₂₃.length, the signs of qdr_nd.angle₁ and qdr_nd.angle₃ are equal, then qdr_nd is a Parallelogram_nd. -/
+theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign (h₁ : qdr_nd.edgeND₁₂.length = qdr_nd.edgeND₃₄.length) (h₂ : qdr_nd.edgeND₄₁.reverse.length = qdr_nd.edgeND₂₃.length) (h : (qdr_nd.angle₁.value.IsPos ∧ qdr_nd.angle₃.value.IsPos) ∨ (qdr_nd.angle₁.value.IsNeg ∧ qdr_nd.angle₃.value.IsNeg)) : qdr_nd.IsParallelogram_nd := by
+  have nontriv₄₁₂ : ¬ Collinear qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂ := by sorry
+  have nontriv₂₃₄ : ¬ Collinear qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄ := by sorry
   have nd₂₄ : qdr_nd.point₄ ≠ qdr_nd.point₂ := by sorry
   have tr_nd₁ : TriangleND P := (TRI_nd qdr_nd.point₄ qdr_nd.point₁ qdr_nd.point₂ nontriv₄₁₂)
   have tr_nd₂ : TriangleND P := (TRI_nd qdr_nd.point₂ qdr_nd.point₃ qdr_nd.point₄ nontriv₂₃₄)
@@ -596,8 +495,8 @@ theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign (h₁ : qdr_nd.edg
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, AB = CD, AD = BC, the signs of ∠DAB and ∠BCD are equal, then Quadrilateral ABCD is a Parallelogram_nd. -/
 theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign_variant (h₁ : (SEG A B).length = (SEG C D).length) (h₂ : (SEG A D).length = (SEG B C).length) (h : ((ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value.IsPos ∧ (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value.IsPos) ∨ ((ANG D A B (QDR_nd A B C D nd).nd₁₄.out (QDR_nd A B C D nd).nd₁₂.out).value.IsNeg ∧ (ANG B C D (QDR_nd A B C D nd).nd₂₃.out.symm (QDR_nd A B C D nd).nd₃₄.out).value.IsNeg)) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign (QDR_nd A B C D nd) h₁ h₂ h
 
-/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edge_nd₁₂.length = qdr_nd.edge_nd₃₄.length, qdr_nd.edge_nd₁₄.length = qdr_nd.edge_nd₂₃.length, the signs of qdr_nd.angle₂ and qdr_nd.angle₄ are equal, then qdr_nd is a Parallelogram_nd. -/
-theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign' (h₁ : qdr_nd.edge_nd₁₂.length = qdr_nd.edge_nd₃₄.length) (h₂ : qdr_nd.edge_nd₁₄.length = qdr_nd.edge_nd₂₃.length) (h : (qdr_nd.angle₂.value.IsPos ∧ qdr_nd.angle₄.value.IsPos) ∨ (qdr_nd.angle₂.value.IsNeg ∧ qdr_nd.angle₄.value.IsNeg)) : qdr_nd.IsParallelogram_nd := by sorry
+/-- Given Quadrilateral_nd qdr_nd, qdr_nd.edgeND₁₂.length = qdr_nd.edgeND₃₄.length, qdr_nd.edgeND₄₁.reverse.length = qdr_nd.edgeND₂₃.length, the signs of qdr_nd.angle₂ and qdr_nd.angle₄ are equal, then qdr_nd is a Parallelogram_nd. -/
+theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign' (h₁ : qdr_nd.edgeND₁₂.length = qdr_nd.edgeND₃₄.length) (h₂ : qdr_nd.edgeND₄₁.reverse.length = qdr_nd.edgeND₂₃.length) (h : (qdr_nd.angle₂.value.IsPos ∧ qdr_nd.angle₄.value.IsPos) ∨ (qdr_nd.angle₂.value.IsNeg ∧ qdr_nd.angle₄.value.IsNeg)) : qdr_nd.IsParallelogram_nd := by sorry
 
 /-- Given four points ABCD and Quadrilateral ABCD IsNd, AB = CD, AD = BC, the signs of ∠ABC and ∠CDA are equal, then Quadrilateral ABCD is a Parallelogram_nd. -/
 theorem qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign'_variant (h₁ : (SEG A B).length = (SEG C D).length) (h₂ : (SEG A D).length = (SEG B C).length) (h : ((ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value.IsPos ∧ (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value.IsPos) ∨ ((ANG A B C (QDR_nd A B C D nd).nd₁₂.out.symm (QDR_nd A B C D nd).nd₂₃.out).value.IsNeg ∧ (ANG C D A (QDR_nd A B C D nd).nd₃₄.out.symm (QDR_nd A B C D nd).nd₁₄.out.symm).value.IsNeg)) : (QDR_nd A B C D nd).IsParallelogram_nd := qdr_nd_is_prg_nd_of_eq_length_eq_length_eq_angle_sign' (QDR_nd A B C D nd) h₁ h₂ h
@@ -612,8 +511,8 @@ variable (cvx : (QDR A B C D).IsConvex)
 variable {P : Type _} [EuclideanPlane P] (qdr_nd : Quadrilateral_nd P)
 variable {P : Type _} [EuclideanPlane P] (qdr : Quadrilateral P)
 
-/-- Given Quadrilateral_nd qdr_nd, and qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄ and (qdr_nd.edge_nd₁₂).length = (qdr_nd.edge_nd₃₄).length, and qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃ and (qdr_nd.edge_nd₁₄).length = (qdr_nd.edge_nd₂₃).length, qdr_nd is a Parallelogram. -/
-theorem qdr_nd_is_prg_of_para_eq_length_para_eq_length (h₁ : qdr_nd.edge_nd₁₂ ∥ qdr_nd.edge_nd₃₄) (h₂ : qdr_nd.edge_nd₁₂.length = qdr_nd.edge_nd₃₄.length) (H₁ : qdr_nd.edge_nd₁₄ ∥ qdr_nd.edge_nd₂₃) (h₂ : qdr_nd.edge_nd₁₄.length = qdr_nd.edge_nd₂₃.length): qdr_nd.IsParallelogram := by
+/-- Given Quadrilateral_nd qdr_nd, and qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄ and (qdr_nd.edgeND₁₂).length = (qdr_nd.edgeND₃₄).length, and qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃ and (qdr_nd.edgeND₄₁.reverse).length = (qdr_nd.edgeND₂₃).length, qdr_nd is a Parallelogram. -/
+theorem qdr_nd_is_prg_of_para_eq_length_para_eq_length (h₁ : qdr_nd.edgeND₁₂ ∥ qdr_nd.edgeND₃₄) (h₂ : qdr_nd.edgeND₁₂.length = qdr_nd.edgeND₃₄.length) (H₁ : qdr_nd.edgeND₄₁.reverse ∥ qdr_nd.edgeND₂₃) (h₂ : qdr_nd.edgeND₄₁.reverse.length = qdr_nd.edgeND₂₃.length): qdr_nd.IsParallelogram := by
   sorry
 
 /-- Given four points ABCD and Quadrilateral ABCD IsConvex, and AB ∥ CD and AB = CD, Quadrilateral ABCD is a Parallelogram_nd. -/
@@ -638,8 +537,8 @@ variable (cvx : (QDR_nd A B C D nd).IsConvex)
 variable {P : Type _} [EuclideanPlane P] (qdr_cvx : Quadrilateral_cvx P)
 variable {P : Type _} [EuclideanPlane P] (qdr : Quadrilateral P)
 
-/-- Given Quadrilateral_cvx qdr_cvx, qdr_cvx.edge_nd₁₂ ∥ qdr_cvx.edge_nd₃₄ and qdr_cvx.edge_nd₁₄ ∥ qdr_cvx.edge_nd₂₃, then qdr_cvx is a Parallelogram_nd. -/
-theorem qdr_cvx_is_prg_nd_of_para_para (h₁ : qdr_cvx.edge_nd₁₂ ∥ qdr_cvx.edge_nd₃₄) (h₂ : qdr_cvx.edge_nd₁₄ ∥ qdr_cvx.edge_nd₂₃) : qdr_cvx.IsParallelogram_nd := by
+/-- Given Quadrilateral_cvx qdr_cvx, qdr_cvx.edgeND₁₂ ∥ qdr_cvx.edgeND₃₄ and qdr_cvx.edgeND₄₁.reverse ∥ qdr_cvx.edgeND₂₃, then qdr_cvx is a Parallelogram_nd. -/
+theorem qdr_cvx_is_prg_nd_of_para_para (h₁ : qdr_cvx.edgeND₁₂ ∥ qdr_cvx.edgeND₃₄) (h₂ : qdr_cvx.edgeND₄₁.reverse ∥ qdr_cvx.edgeND₂₃) : qdr_cvx.IsParallelogram_nd := by
   unfold Quadrilateral_nd.IsParallelogram_nd
   constructor
   constructor
@@ -664,11 +563,11 @@ theorem qdr_cvx_is_prg_nd_of_para_para_variant (h₁ : (SEG_nd A B (QDR_cvx A B
   exact (QDR_cvx A B C D nd cvx).not_collinear₃₄₁
   exact (QDR_cvx A B C D nd cvx).not_collinear₄₁₂
 
-/-- Given Quadrilateral_cvx qdr_cvx, and (qdr_cvx.edge_nd₁₂).length = (qdr_cvx.edge_nd₃₄).length and qdr_cvx.edge_nd₁₄.length = qdr_cvx.edge_nd₂₃.length, qdr_cvx is a Parallelogram_nd. -/
-theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length (h₁ : qdr_cvx.edge_nd₁₂.length = qdr_cvx.edge_nd₃₄.length) (h₂ : qdr_cvx.edge_nd₁₄.length = qdr_cvx.edge_nd₂₃.length) : qdr_cvx.IsParallelogram_nd := by
+/-- Given Quadrilateral_cvx qdr_cvx, and (qdr_cvx.edgeND₁₂).length = (qdr_cvx.edgeND₃₄).length and qdr_cvx.edgeND₄₁.reverse.length = qdr_cvx.edgeND₂₃.length, qdr_cvx is a Parallelogram_nd. -/
+theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length (h₁ : qdr_cvx.edgeND₁₂.length = qdr_cvx.edgeND₃₄.length) (h₂ : qdr_cvx.edgeND₄₁.reverse.length = qdr_cvx.edgeND₂₃.length) : qdr_cvx.IsParallelogram_nd := by
   unfold Quadrilateral_nd.IsParallelogram_nd
   constructor
-  have heq₁: qdr_cvx.edge_nd₁₄.length = qdr_cvx.edge_nd₁₄.reverse.length := qdr_cvx.edge_nd₁₄.reverse.length_of_rev_eq_length
+  have heq₁: qdr_cvx.edgeND₄₁.reverse.length = qdr_cvx.edgeND₄₁.length := qdr_cvx.edgeND₄₁.length_of_rev_eq_length
   have eq₁: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.not_collinear₄₁₂).edge₁.length = (TRI_nd qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.not_collinear₂₃₄).edge₁.length := h₁
   have eq₂: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.not_collinear₄₁₂).edge₂.length = (TRI_nd qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.not_collinear₂₃₄).edge₂.length := (TRI_nd qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.not_collinear₂₃₄).edge₂.length_of_rev_eq_length
   have eq₃: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.not_collinear₄₁₂).edge₃.length = (TRI_nd qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.not_collinear₂₃₄).edge₃.length := by
@@ -709,24 +608,24 @@ theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length (h₁ : qdr_cvx.edge_nd₁₂.l
     right
     exact near₂
   have very_close₂: qdr_cvx.triangle_nd₁.angle₃.end_ray ∥ qdr_cvx.triangle_nd₃.angle₃.end_ray := very_near₂
-  have close₁: qdr_cvx.edge_nd₁₄.toProj = qdr_cvx.triangle_nd₁.angle₁.start_ray.toProj := by
-    have third: qdr_cvx.edge_nd₁₄.reverse.toProj = qdr_cvx.edge_nd₁₄.toProj := qdr_cvx.edge_nd₁₄.toProj_of_rev_eq_toProj
+  have close₁: qdr_cvx.edgeND₄₁.reverse.toProj = qdr_cvx.triangle_nd₁.angle₁.start_ray.toProj := by
+    have third: qdr_cvx.edgeND₄₁.toProj = qdr_cvx.edgeND₄₁.reverse.toProj := qdr_cvx.edgeND₄₁.reverse.toProj_of_rev_eq_toProj
     exact id third.symm
-  have close₂: qdr_cvx.edge_nd₂₃.toProj = qdr_cvx.triangle_nd₃.angle₁.start_ray.toProj := by rfl
-  have close₃: qdr_cvx.edge_nd₁₂.toProj = qdr_cvx.triangle_nd₁.angle₃.end_ray.toProj := by
-    have last: qdr_cvx.edge_nd₁₂.reverse.toRay = qdr_cvx.triangle_nd₁.angle₃.end_ray := by rfl
-    have second: qdr_cvx.edge_nd₁₂.reverse.toProj = qdr_cvx.triangle_nd₁.angle₃.end_ray.toProj := by rfl
-    have third: qdr_cvx.edge_nd₁₂.reverse.toProj = qdr_cvx.edge_nd₁₂.toProj := qdr_cvx.edge_nd₁₂.toProj_of_rev_eq_toProj
+  have close₂: qdr_cvx.edgeND₂₃.toProj = qdr_cvx.triangle_nd₃.angle₁.start_ray.toProj := by rfl
+  have close₃: qdr_cvx.edgeND₁₂.toProj = qdr_cvx.triangle_nd₁.angle₃.end_ray.toProj := by
+    have last: qdr_cvx.edgeND₁₂.reverse.toRay = qdr_cvx.triangle_nd₁.angle₃.end_ray := by rfl
+    have second: qdr_cvx.edgeND₁₂.reverse.toProj = qdr_cvx.triangle_nd₁.angle₃.end_ray.toProj := by rfl
+    have third: qdr_cvx.edgeND₁₂.reverse.toProj = qdr_cvx.edgeND₁₂.toProj := qdr_cvx.edgeND₁₂.toProj_of_rev_eq_toProj
     rw [third.symm, second, last.symm]
-  have close₄: qdr_cvx.edge_nd₃₄.toProj = qdr_cvx.triangle_nd₃.angle₃.end_ray.toProj := by
-    have last: qdr_cvx.edge_nd₃₄.reverse.toRay = qdr_cvx.triangle_nd₃.angle₃.end_ray := by rfl
-    have second: qdr_cvx.edge_nd₃₄.reverse.toProj = qdr_cvx.triangle_nd₃.angle₃.end_ray.toProj := by rfl
-    have third: qdr_cvx.edge_nd₃₄.reverse.toProj = qdr_cvx.edge_nd₃₄.toProj := qdr_cvx.edge_nd₃₄.toProj_of_rev_eq_toProj
+  have close₄: qdr_cvx.edgeND₃₄.toProj = qdr_cvx.triangle_nd₃.angle₃.end_ray.toProj := by
+    have last: qdr_cvx.edgeND₃₄.reverse.toRay = qdr_cvx.triangle_nd₃.angle₃.end_ray := by rfl
+    have second: qdr_cvx.edgeND₃₄.reverse.toProj = qdr_cvx.triangle_nd₃.angle₃.end_ray.toProj := by rfl
+    have third: qdr_cvx.edgeND₃₄.reverse.toProj = qdr_cvx.edgeND₃₄.toProj := qdr_cvx.edgeND₃₄.toProj_of_rev_eq_toProj
     rw [third.symm, second, last.symm]
-  have win₁: qdr_cvx.edge_nd₁₂.toProj = qdr_cvx.edge_nd₃₄.toProj := by
+  have win₁: qdr_cvx.edgeND₁₂.toProj = qdr_cvx.edgeND₃₄.toProj := by
     rw [close₃, close₄]
     exact very_close₂
-  have win₂: qdr_cvx.edge_nd₁₄.toProj = qdr_cvx.edge_nd₂₃.toProj := by
+  have win₂: qdr_cvx.edgeND₄₁.reverse.toProj = qdr_cvx.edgeND₂₃.toProj := by
     rw [close₁, close₂]
     exact very_close₁
   exact ⟨win₁, win₂⟩
@@ -740,7 +639,7 @@ theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length (h₁ : qdr_cvx.edge_nd₁₂.l
 theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length_variant (h₁ : (SEG A B).length = (SEG C D).length) (h₂ : (SEG A D).length = (SEG B C).length) : (Quadrilateral_nd.mk_is_nd nd) IsParallelogram_nd := by
   unfold Quadrilateral_nd.IsParallelogram_nd
   constructor
-  have heq₁: (SEG A D).length = (QDR_cvx A B C D nd cvx).edge_nd₁₄.reverse.length := (QDR_cvx A B C D nd cvx).edge_nd₁₄.reverse.length_of_rev_eq_length
+  have heq₁: (SEG A D).length = (QDR_cvx A B C D nd cvx).edgeND₄₁.length := (QDR_cvx A B C D nd cvx).edgeND₄₁.length_of_rev_eq_length
   have eq₁: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).not_collinear₄₁₂).edge₁.length = (TRI_nd (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₂₃₄).edge₁.length := h₁
   have eq₂: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).not_collinear₄₁₂).edge₂.length = (TRI_nd (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₂₃₄).edge₂.length := (TRI_nd (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₂₃₄).edge₂.length_of_rev_eq_length
   have eq₃: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).not_collinear₄₁₂).edge₃.length = (TRI_nd (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₂₃₄).edge₃.length := by
@@ -781,24 +680,24 @@ theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length_variant (h₁ : (SEG A B).lengt
     right
     exact near₂
   have very_close₂: (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray ∥ (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray := very_near₂
-  have close₁: (QDR_cvx A B C D nd cvx).edge_nd₁₄.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₁.start_ray.toProj := by
-    have third: (QDR_cvx A B C D nd cvx).edge_nd₁₄.reverse.toProj = (QDR_cvx A B C D nd cvx).edge_nd₁₄.toProj := (QDR_cvx A B C D nd cvx).edge_nd₁₄.toProj_of_rev_eq_toProj
+  have close₁: (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₁.start_ray.toProj := by
+    have third: (QDR_cvx A B C D nd cvx).edgeND₄₁.toProj = (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toProj := (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toProj_of_rev_eq_toProj
     exact id third.symm
-  have close₂: (QDR_cvx A B C D nd cvx).edge_nd₂₃.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₁.start_ray.toProj := by rfl
-  have close₃: (QDR_cvx A B C D nd cvx).edge_nd₁₂.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray.toProj := by
-    have last: (QDR_cvx A B C D nd cvx).edge_nd₁₂.reverse.toRay = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray := by rfl
-    have second: (QDR_cvx A B C D nd cvx).edge_nd₁₂.reverse.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray.toProj := by rfl
-    have third: (QDR_cvx A B C D nd cvx).edge_nd₁₂.reverse.toProj = (QDR_cvx A B C D nd cvx).edge_nd₁₂.toProj := (QDR_cvx A B C D nd cvx).edge_nd₁₂.toProj_of_rev_eq_toProj
+  have close₂: (QDR_cvx A B C D nd cvx).edgeND₂₃.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₁.start_ray.toProj := by rfl
+  have close₃: (QDR_cvx A B C D nd cvx).edgeND₁₂.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray.toProj := by
+    have last: (QDR_cvx A B C D nd cvx).edgeND₁₂.reverse.toRay = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray := by rfl
+    have second: (QDR_cvx A B C D nd cvx).edgeND₁₂.reverse.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₁.angle₃.end_ray.toProj := by rfl
+    have third: (QDR_cvx A B C D nd cvx).edgeND₁₂.reverse.toProj = (QDR_cvx A B C D nd cvx).edgeND₁₂.toProj := (QDR_cvx A B C D nd cvx).edgeND₁₂.toProj_of_rev_eq_toProj
     rw [third.symm, second, last.symm]
-  have close₄: (QDR_cvx A B C D nd cvx).edge_nd₃₄.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray.toProj := by
-    have last: (QDR_cvx A B C D nd cvx).edge_nd₃₄.reverse.toRay = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray := by rfl
-    have second: (QDR_cvx A B C D nd cvx).edge_nd₃₄.reverse.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray.toProj := by rfl
-    have third: (QDR_cvx A B C D nd cvx).edge_nd₃₄.reverse.toProj = (QDR_cvx A B C D nd cvx).edge_nd₃₄.toProj := (QDR_cvx A B C D nd cvx).edge_nd₃₄.toProj_of_rev_eq_toProj
+  have close₄: (QDR_cvx A B C D nd cvx).edgeND₃₄.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray.toProj := by
+    have last: (QDR_cvx A B C D nd cvx).edgeND₃₄.reverse.toRay = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray := by rfl
+    have second: (QDR_cvx A B C D nd cvx).edgeND₃₄.reverse.toProj = (QDR_cvx A B C D nd cvx).triangle_nd₃.angle₃.end_ray.toProj := by rfl
+    have third: (QDR_cvx A B C D nd cvx).edgeND₃₄.reverse.toProj = (QDR_cvx A B C D nd cvx).edgeND₃₄.toProj := (QDR_cvx A B C D nd cvx).edgeND₃₄.toProj_of_rev_eq_toProj
     rw [third.symm, second, last.symm]
-  have win₁: (QDR_cvx A B C D nd cvx).edge_nd₁₂.toProj = (QDR_cvx A B C D nd cvx).edge_nd₃₄.toProj := by
+  have win₁: (QDR_cvx A B C D nd cvx).edgeND₁₂.toProj = (QDR_cvx A B C D nd cvx).edgeND₃₄.toProj := by
     rw [close₃, close₄]
     exact very_close₂
-  have win₂: (QDR_cvx A B C D nd cvx).edge_nd₁₄.toProj = (QDR_cvx A B C D nd cvx).edge_nd₂₃.toProj := by
+  have win₂: (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toProj = (QDR_cvx A B C D nd cvx).edgeND₂₃.toProj := by
     rw [close₁, close₂]
     exact very_close₁
   exact ⟨win₁, win₂⟩
@@ -808,12 +707,12 @@ theorem qdr_cvx_is_prg_nd_of_eq_length_eq_length_variant (h₁ : (SEG A B).lengt
   exact (QDR_cvx A B C D nd cvx).not_collinear₃₄₁
   exact (QDR_cvx A B C D nd cvx).not_collinear₄₁₂
 
-/-- Given Quadrilateral_cvx qdr_cvx, and qdr_cvx.edge_nd₁₂ ∥ qdr_cvx.edge_nd₃₄ and (qdr_cvx.edge_nd₁₂).length = (qdr_cvx.edge_nd₃₄).length, qdr_cvx is a Parallelogram_nd. -/
-theorem qdr_cvx_is_prg_nd_of_para_eq_length (h₁ : qdr_cvx.edge_nd₁₂ ∥ qdr_cvx.edge_nd₃₄) (h₂ : qdr_cvx.edge_nd₁₂.length = qdr_cvx.edge_nd₃₄.length) : qdr_cvx.IsParallelogram_nd := by
+/-- Given Quadrilateral_cvx qdr_cvx, and qdr_cvx.edgeND₁₂ ∥ qdr_cvx.edgeND₃₄ and (qdr_cvx.edgeND₁₂).length = (qdr_cvx.edgeND₃₄).length, qdr_cvx is a Parallelogram_nd. -/
+theorem qdr_cvx_is_prg_nd_of_para_eq_length (h₁ : qdr_cvx.edgeND₁₂ ∥ qdr_cvx.edgeND₃₄) (h₂ : qdr_cvx.edgeND₁₂.length = qdr_cvx.edgeND₃₄.length) : qdr_cvx.IsParallelogram_nd := by
   unfold Quadrilateral_nd.IsParallelogram_nd
   constructor
-  unfold parallel at h₁
-  have h: qdr_cvx.edge_nd₁₂.toDir = qdr_cvx.edge_nd₃₄.toDir ∨ qdr_cvx.edge_nd₁₂.toDir = - qdr_cvx.edge_nd₃₄.toDir := by
+  unfold Parallel at h₁
+  have h: qdr_cvx.edgeND₁₂.toDir = qdr_cvx.edgeND₃₄.toDir ∨ qdr_cvx.edgeND₁₂.toDir = - qdr_cvx.edgeND₃₄.toDir := by
     apply Dir.toProj_eq_toProj_iff.1
     exact h₁
   have diag_ng_para: ¬ qdr_cvx.diag_nd₁₃.toProj = qdr_cvx.diag_nd₂₄.toProj := qdr_cvx.diag_not_para
@@ -821,15 +720,15 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length (h₁ : qdr_cvx.edge_nd₁₂ ∥ qd
   -- Case that is not convex, goal is prove contra
   have angle₁_eq_angle₄: (ANG qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁).value = (ANG qdr_cvx.point₂ qdr_cvx.point₁ qdr_cvx.point₄).value := by
     apply ang_eq_ang_of_toDir_rev_toDir
-    have ray₁₂_toDir_eq_SegND₁₂_toDir: qdr_cvx.edge_nd₁₂.toDir = (ANG qdr_cvx.point₂ qdr_cvx.point₁ qdr_cvx.point₄).start_ray.toDir := by rfl
-    have ray₄₃_toDir_eq_SegND₃₄_rev_toDir: qdr_cvx.edge_nd₃₄.reverse.toDir = (ANG qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁).start_ray.toDir := by rfl
-    have SegND₄₃_toDir_neg_SegND₃₄_rev_toDir: qdr_cvx.edge_nd₃₄.reverse.toDir = - qdr_cvx.edge_nd₃₄.toDir := by apply SegND.toDir_of_rev_eq_neg_toDir
+    have ray₁₂_toDir_eq_SegND₁₂_toDir: qdr_cvx.edgeND₁₂.toDir = (ANG qdr_cvx.point₂ qdr_cvx.point₁ qdr_cvx.point₄).start_ray.toDir := by rfl
+    have ray₄₃_toDir_eq_SegND₃₄_rev_toDir: qdr_cvx.edgeND₃₄.reverse.toDir = (ANG qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁).start_ray.toDir := by rfl
+    have SegND₄₃_toDir_neg_SegND₃₄_rev_toDir: qdr_cvx.edgeND₃₄.reverse.toDir = - qdr_cvx.edgeND₃₄.toDir := by apply SegND.toDir_of_rev_eq_neg_toDir
     rw [ray₁₂_toDir_eq_SegND₁₂_toDir.symm, ray₄₃_toDir_eq_SegND₃₄_rev_toDir.symm, SegND₄₃_toDir_neg_SegND₃₄_rev_toDir, case_not_convex]
-    exact qdr_cvx.edge_nd₁₄.toDir_of_rev_eq_neg_toDir
+    exact qdr_cvx.edgeND₄₁.reverse.toDir_of_rev_eq_neg_toDir
   have IsCongrTo₁₄: TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁ IsCongrTo TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄ := by
     have edge₂_eq: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).1.edge₂.length = (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄).1.edge₂.length := by apply length_of_rev_eq_length'
     have edge₃_eq: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).1.edge₃.length = (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄).1.edge₃.length := by
-      have eq_length: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).1.edge₃.length = qdr_cvx.edge_nd₃₄.length := by apply length_of_rev_eq_length'
+      have eq_length: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).1.edge₃.length = qdr_cvx.edgeND₃₄.length := by apply length_of_rev_eq_length'
       rw [eq_length]
       exact h₂.symm
     apply TriangleND.congr_of_SAS edge₂_eq angle₁_eq_angle₄ edge₃_eq
@@ -841,9 +740,9 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length (h₁ : qdr_cvx.edge_nd₁₂ ∥ qd
   -- Use angle_eq to prove two diag para.
   have prepa₂: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.start_ray.reverse.toDir = (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄).angle₃.start_ray.toDir := by
     have prepaA: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.start_ray.reverse.toDir = - (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.start_ray.toDir := (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.start_ray.toDir_of_rev_eq_neg_toDir
-    have prepaB: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.start_ray.toDir = qdr_cvx.edge_nd₁₄.toDir := by rfl
-    have prepaC: (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄).angle₃.start_ray.toDir = qdr_cvx.edge_nd₁₄.reverse.toDir := by rfl
-    have prepaD: qdr_cvx.edge_nd₁₄.reverse.toDir = - qdr_cvx.edge_nd₁₄.toDir := qdr_cvx.edge_nd₁₄.toDir_of_rev_eq_neg_toDir
+    have prepaB: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.start_ray.toDir = qdr_cvx.edgeND₄₁.reverse.toDir := by rfl
+    have prepaC: (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄).angle₃.start_ray.toDir = qdr_cvx.edgeND₄₁.toDir := by rfl
+    have prepaD: qdr_cvx.edgeND₄₁.toDir = - qdr_cvx.edgeND₄₁.reverse.toDir := qdr_cvx.edgeND₄₁.reverse.toDir_of_rev_eq_neg_toDir
     rw [prepaA, prepaB, prepaC, prepaD]
   have prepa₃: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.end_ray.reverse.toDir = (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ qdr_cvx.not_collinear₁₂₄).angle₃.end_ray.toDir := (end_ray_toDir_rev_toDir_of_ang_rev_ang prepa₂.symm prepa₁).symm
   have prepa₄: (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.end_ray.reverse.toDir = - (TRI_nd qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ qdr_cvx.not_collinear₄₃₁).angle₃.end_ray.toDir := by
@@ -886,12 +785,12 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length (h₁ : qdr_cvx.edge_nd₁₂ ∥ qd
     exact K₆
   have near: (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.not_collinear₁₂₃).angle₃.end_ray.toDir = (TRI_nd qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.not_collinear₃₄₁).angle₃.end_ray.reverse.toDir := end_ray_toDir_rev_toDir_of_ang_rev_ang pr₂ pr₁
   have J₁: (TRI_nd qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.not_collinear₃₄₁).angle₃.end_ray.reverse.toDir = - (TRI_nd qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.not_collinear₃₄₁).angle₃.end_ray.toDir := (TRI_nd qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.not_collinear₃₄₁).angle₃.end_ray.toDir_of_rev_eq_neg_toDir
-  have J₂: (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.not_collinear₁₂₃).angle₃.end_ray.toDir = qdr_cvx.edge_nd₂₃.reverse.toDir := by rfl
-  have J₃: (TRI_nd qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.not_collinear₃₄₁).angle₃.end_ray.toDir = qdr_cvx.edge_nd₁₄.toDir := by rfl
-  have J₄: qdr_cvx.edge_nd₂₃.reverse.toDir = - qdr_cvx.edge_nd₂₃.toDir := qdr_cvx.edge_nd₂₃.toDir_of_rev_eq_neg_toDir
+  have J₂: (TRI_nd qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.not_collinear₁₂₃).angle₃.end_ray.toDir = qdr_cvx.edgeND₂₃.reverse.toDir := by rfl
+  have J₃: (TRI_nd qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.not_collinear₃₄₁).angle₃.end_ray.toDir = qdr_cvx.edgeND₄₁.reverse.toDir := by rfl
+  have J₄: qdr_cvx.edgeND₂₃.reverse.toDir = - qdr_cvx.edgeND₂₃.toDir := qdr_cvx.edgeND₂₃.toDir_of_rev_eq_neg_toDir
   rw [J₁, J₂, J₃, J₄] at near
   simp only [neg_inj] at near
-  have close: qdr_cvx.edge_nd₂₃.toProj = qdr_cvx.edge_nd₁₄.toProj := by
+  have close: qdr_cvx.edgeND₂₃.toProj = qdr_cvx.edgeND₄₁.reverse.toProj := by
     apply Dir.toProj_eq_toProj_iff.mpr
     left
     exact near
@@ -909,8 +808,8 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length (h₁ : qdr_cvx.edge_nd₁₂ ∥ qd
 theorem qdr_cvx_is_prg_nd_of_para_eq_length_variant (h₁ : (SEG_nd A B (QDR_cvx A B C D nd cvx).nd₁₂.out) ∥ (SEG_nd C D (QDR_cvx A B C D nd cvx).nd₃₄.out)) (h₂ : (SEG_nd A B (QDR_cvx A B C D nd cvx).nd₁₂.out).length = (SEG_nd C D (QDR_cvx A B C D nd cvx).nd₃₄.out).length) : (Quadrilateral_nd.mk_is_nd nd) IsParallelogram_nd := by
    unfold Quadrilateral_nd.IsParallelogram_nd
    constructor
-   unfold parallel at h₁
-   have h: (QDR_cvx A B C D nd cvx).edge_nd₁₂.toDir = (QDR_cvx A B C D nd cvx).edge_nd₃₄.toDir ∨ (QDR_cvx A B C D nd cvx).edge_nd₁₂.toDir = - (QDR_cvx A B C D nd cvx).edge_nd₃₄.toDir := by
+   unfold Parallel at h₁
+   have h: (QDR_cvx A B C D nd cvx).edgeND₁₂.toDir = (QDR_cvx A B C D nd cvx).edgeND₃₄.toDir ∨ (QDR_cvx A B C D nd cvx).edgeND₁₂.toDir = - (QDR_cvx A B C D nd cvx).edgeND₃₄.toDir := by
      apply Dir.toProj_eq_toProj_iff.1
      exact h₁
    have diag_ng_para: ¬ (QDR_cvx A B C D nd cvx).diag_nd₁₃.toProj = (QDR_cvx A B C D nd cvx).diag_nd₂₄.toProj := (QDR_cvx A B C D nd cvx).diag_not_para
@@ -918,15 +817,15 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length_variant (h₁ : (SEG_nd A B (QDR_cvx
    -- Case that is not convex, goal is prove contra
    have angle₁_eq_angle₄: (ANG (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).nd₃₄.out.symm (QDR_cvx A B C D nd cvx).nd₁₄.out.symm).value = (ANG (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).nd₁₂.out (QDR_cvx A B C D nd cvx).nd₁₄.out).value := by
      apply ang_eq_ang_of_toDir_rev_toDir
-     have ray₁₂_toDir_eq_Seg_nd₁₂_toDir: (QDR_cvx A B C D nd cvx).edge_nd₁₂.toDir = (ANG (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).nd₁₂.out (QDR_cvx A B C D nd cvx).nd₁₄.out).start_ray.toDir := by rfl
-     have ray₄₃_toDir_eq_Seg_nd₃₄_rev_toDir: (QDR_cvx A B C D nd cvx).edge_nd₃₄.reverse.toDir = (ANG (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).nd₃₄.out.symm (QDR_cvx A B C D nd cvx).nd₁₄.out.symm).start_ray.toDir := by rfl
-     have Seg_nd₄₃_toDir_neg_Seg_nd₃₄_rev_toDir: (QDR_cvx A B C D nd cvx).edge_nd₃₄.reverse.toDir = - (QDR_cvx A B C D nd cvx).edge_nd₃₄.toDir := by apply SegND.toDir_of_rev_eq_neg_toDir
+     have ray₁₂_toDir_eq_Seg_nd₁₂_toDir: (QDR_cvx A B C D nd cvx).edgeND₁₂.toDir = (ANG (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).nd₁₂.out (QDR_cvx A B C D nd cvx).nd₁₄.out).start_ray.toDir := by rfl
+     have ray₄₃_toDir_eq_Seg_nd₃₄_rev_toDir: (QDR_cvx A B C D nd cvx).edgeND₃₄.reverse.toDir = (ANG (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).nd₃₄.out.symm (QDR_cvx A B C D nd cvx).nd₁₄.out.symm).start_ray.toDir := by rfl
+     have Seg_nd₄₃_toDir_neg_Seg_nd₃₄_rev_toDir: (QDR_cvx A B C D nd cvx).edgeND₃₄.reverse.toDir = - (QDR_cvx A B C D nd cvx).edgeND₃₄.toDir := by apply SegND.toDir_of_rev_eq_neg_toDir
      rw [ray₁₂_toDir_eq_Seg_nd₁₂_toDir.symm, ray₄₃_toDir_eq_Seg_nd₃₄_rev_toDir.symm, Seg_nd₄₃_toDir_neg_Seg_nd₃₄_rev_toDir, case_not_convex]
-     exact (QDR_cvx A B C D nd cvx).edge_nd₁₄.toDir_of_rev_eq_neg_toDir
+     exact (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toDir_of_rev_eq_neg_toDir
    have IsCongrTo₁₄: TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁ IsCongrTo TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄ := by
      have edge₂_eq: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).1.edge₂.length = (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄).1.edge₂.length := by apply length_of_rev_eq_length'
      have edge₃_eq: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).1.edge₃.length = (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄).1.edge₃.length := by
-       have eq_length: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).1.edge₃.length = (QDR_cvx A B C D nd cvx).edge_nd₃₄.length := by apply length_of_rev_eq_length'
+       have eq_length: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).1.edge₃.length = (QDR_cvx A B C D nd cvx).edgeND₃₄.length := by apply length_of_rev_eq_length'
        rw [eq_length]
        exact h₂.symm
      apply TriangleND.congr_of_SAS edge₂_eq angle₁_eq_angle₄ edge₃_eq
@@ -938,9 +837,9 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length_variant (h₁ : (SEG_nd A B (QDR_cvx
    -- Use angle_eq to prove two diag para.
    have prepa₂: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.start_ray.reverse.toDir = (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄).angle₃.start_ray.toDir := by
      have prepaA: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.start_ray.reverse.toDir = - (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.start_ray.toDir := (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.start_ray.toDir_of_rev_eq_neg_toDir
-     have prepaB: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.start_ray.toDir = (QDR_cvx A B C D nd cvx).edge_nd₁₄.toDir := by rfl
-     have prepaC: (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄).angle₃.start_ray.toDir = (QDR_cvx A B C D nd cvx).edge_nd₁₄.reverse.toDir := by rfl
-     have prepaD: (QDR_cvx A B C D nd cvx).edge_nd₁₄.reverse.toDir = - (QDR_cvx A B C D nd cvx).edge_nd₁₄.toDir := (QDR_cvx A B C D nd cvx).edge_nd₁₄.toDir_of_rev_eq_neg_toDir
+     have prepaB: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.start_ray.toDir = (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toDir := by rfl
+     have prepaC: (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄).angle₃.start_ray.toDir = (QDR_cvx A B C D nd cvx).edgeND₄₁.toDir := by rfl
+     have prepaD: (QDR_cvx A B C D nd cvx).edgeND₄₁.toDir = - (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toDir := (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toDir_of_rev_eq_neg_toDir
      rw [prepaA, prepaB, prepaC, prepaD]
    have prepa₃: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.end_ray.reverse.toDir = (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).not_collinear₁₂₄).angle₃.end_ray.toDir := (end_ray_toDir_rev_toDir_of_ang_rev_ang prepa₂.symm prepa₁).symm
    have prepa₄: (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.end_ray.reverse.toDir = - (TRI_nd (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₄₃₁).angle₃.end_ray.toDir := by
@@ -983,12 +882,12 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length_variant (h₁ : (SEG_nd A B (QDR_cvx
      exact K₆
    have near: (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂ (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).not_collinear₁₂₃).angle₃.end_ray.toDir = (TRI_nd (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₃₄₁).angle₃.end_ray.reverse.toDir := end_ray_toDir_rev_toDir_of_ang_rev_ang pr₂ pr₁
    have J₁: (TRI_nd (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₃₄₁).angle₃.end_ray.reverse.toDir = - (TRI_nd (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₃₄₁).angle₃.end_ray.toDir := (TRI_nd (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₃₄₁).angle₃.end_ray.toDir_of_rev_eq_neg_toDir
-   have J₂: (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂(QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).not_collinear₁₂₃).angle₃.end_ray.toDir = (QDR_cvx A B C D nd cvx).edge_nd₂₃.reverse.toDir := by rfl
-   have J₃: (TRI_nd (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₃₄₁).angle₃.end_ray.toDir = (QDR_cvx A B C D nd cvx).edge_nd₁₄.toDir := by rfl
-   have J₄: (QDR_cvx A B C D nd cvx).edge_nd₂₃.reverse.toDir = - (QDR_cvx A B C D nd cvx).edge_nd₂₃.toDir := (QDR_cvx A B C D nd cvx).edge_nd₂₃.toDir_of_rev_eq_neg_toDir
+   have J₂: (TRI_nd (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).point₂(QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).not_collinear₁₂₃).angle₃.end_ray.toDir = (QDR_cvx A B C D nd cvx).edgeND₂₃.reverse.toDir := by rfl
+   have J₃: (TRI_nd (QDR_cvx A B C D nd cvx).point₃ (QDR_cvx A B C D nd cvx).point₄ (QDR_cvx A B C D nd cvx).point₁ (QDR_cvx A B C D nd cvx).not_collinear₃₄₁).angle₃.end_ray.toDir = (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toDir := by rfl
+   have J₄: (QDR_cvx A B C D nd cvx).edgeND₂₃.reverse.toDir = - (QDR_cvx A B C D nd cvx).edgeND₂₃.toDir := (QDR_cvx A B C D nd cvx).edgeND₂₃.toDir_of_rev_eq_neg_toDir
    rw [J₁, J₂, J₃, J₄] at near
    simp only [neg_inj] at near
-   have close: (QDR_cvx A B C D nd cvx).edge_nd₂₃.toProj = (QDR_cvx A B C D nd cvx).edge_nd₁₄.toProj := by
+   have close: (QDR_cvx A B C D nd cvx).edgeND₂₃.toProj = (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toProj := by
      apply Dir.toProj_eq_toProj_iff.mpr
      left
      exact near
@@ -1003,25 +902,25 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length_variant (h₁ : (SEG_nd A B (QDR_cvx
    exact (QDR_cvx A B C D nd cvx).not_collinear₄₁₂
 
 
-/-- Given Quadrilateral_cvx qdr_cvx, and qdr_cvx.edge_nd₁₄ ∥ qdr_cvx.edge_nd₂₃ and (qdr_cvx.edge_nd₁₄).length = (qdr_cvx.edge_nd₂₃).length, qdr_cvx is a Parallelogram_nd. -/
-theorem qdr_cvx_is_prg_nd_of_para_eq_length' (h₁ : qdr_cvx.edge_nd₁₄ ∥ qdr_cvx.edge_nd₂₃) (h₂ : qdr_cvx.edge_nd₁₄.length = qdr_cvx.edge_nd₂₃.length) : qdr_cvx.IsParallelogram_nd := by
+/-- Given Quadrilateral_cvx qdr_cvx, and qdr_cvx.edgeND₄₁.reverse ∥ qdr_cvx.edgeND₂₃ and (qdr_cvx.edgeND₄₁.reverse).length = (qdr_cvx.edgeND₂₃).length, qdr_cvx is a Parallelogram_nd. -/
+theorem qdr_cvx_is_prg_nd_of_para_eq_length' (h₁ : qdr_cvx.edgeND₄₁.reverse ∥ qdr_cvx.edgeND₂₃) (h₂ : qdr_cvx.edgeND₄₁.reverse.length = qdr_cvx.edgeND₂₃.length) : qdr_cvx.IsParallelogram_nd := by
   unfold Quadrilateral_nd.IsParallelogram_nd
   constructor
   let permute_convex := Quadrilateral_cvx.mk_is_convex qdr_cvx.permute_is_convex
-  have K₁: permute_convex.edge_nd₁₂.toProj = qdr_cvx.edge_nd₂₃.toProj := by rfl
-  have K₂: permute_convex.edge_nd₁₂.length = qdr_cvx.edge_nd₂₃.length := by rfl
-  have j₂: permute_convex.edge_nd₃₄ = qdr_cvx.edge_nd₁₄.reverse := by rfl
-  have K₃: permute_convex.edge_nd₃₄.toProj = qdr_cvx.edge_nd₁₄.toProj := by
+  have K₁: permute_convex.edgeND₁₂.toProj = qdr_cvx.edgeND₂₃.toProj := by rfl
+  have K₂: permute_convex.edgeND₁₂.length = qdr_cvx.edgeND₂₃.length := by rfl
+  have j₂: permute_convex.edgeND₃₄ = qdr_cvx.edgeND₄₁ := by rfl
+  have K₃: permute_convex.edgeND₃₄.toProj = qdr_cvx.edgeND₄₁.reverse.toProj := by
     rw [j₂]
-    apply SegND.toProj_of_rev_eq_toProj
-  have K₄: permute_convex.edge_nd₃₄.length = qdr_cvx.edge_nd₁₄.length := by
+    apply SegND.toProj_of_rev_eq_toProj.symm
+  have K₄: permute_convex.edgeND₃₄.length = qdr_cvx.edgeND₄₁.reverse.length := by
     rw [j₂]
-    apply SegND.length_of_rev_eq_length
-  have H₁: permute_convex.edge_nd₁₂.toProj = permute_convex.edge_nd₃₄.toProj := by
+    apply SegND.length_of_rev_eq_length.symm
+  have H₁: permute_convex.edgeND₁₂.toProj = permute_convex.edgeND₃₄.toProj := by
     rw [K₁, K₃]
-    unfold parallel at h₁
+    unfold Parallel at h₁
     exact h₁.symm
-  have H₂: permute_convex.edge_nd₁₂.length = permute_convex.edge_nd₃₄.length := by
+  have H₂: permute_convex.edgeND₁₂.length = permute_convex.edgeND₃₄.length := by
     rw [K₂, K₄]
     exact h₂.symm
   have H: permute_convex.IsParallelogram_nd := qdr_cvx_is_prg_nd_of_para_eq_length permute_convex H₁ H₂
@@ -1030,15 +929,15 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length' (h₁ : qdr_cvx.edge_nd₁₄ ∥ q
   unfold Quadrilateral_nd.IsParallelogram_para
   unfold Quadrilateral_nd.IsParallelogram_para at para
   rcases para with ⟨para₁, para₂⟩
-  unfold parallel
-  have K₉: permute_convex.edge_nd₁₄.reverse.toProj = qdr_cvx.edge_nd₁₂.toProj := by rfl
-  have K₁₀: permute_convex.edge_nd₁₄.reverse.toProj = permute_convex.edge_nd₁₄.toProj := by apply SegND.toProj_of_rev_eq_toProj
-  have K₈: permute_convex.edge_nd₂₃.toProj = qdr_cvx.edge_nd₃₄.toProj := by rfl
+  unfold Parallel
+  have K₉: permute_convex.edgeND₄₁.toProj = qdr_cvx.edgeND₁₂.toProj := by rfl
+  have K₁₀: permute_convex.edgeND₄₁.toProj = permute_convex.edgeND₄₁.reverse.toProj := by apply SegND.toProj_of_rev_eq_toProj.symm
+  have K₈: permute_convex.edgeND₂₃.toProj = qdr_cvx.edgeND₃₄.toProj := by rfl
   rw [K₁₀] at K₉
-  unfold parallel at para₁
-  unfold parallel at para₂
+  unfold Parallel at para₁
+  unfold Parallel at para₂
   constructor
-  have key: qdr_cvx.edge_nd₁₂.toProj = qdr_cvx.edge_nd₃₄.toProj := by
+  have key: qdr_cvx.edgeND₁₂.toProj = qdr_cvx.edgeND₃₄.toProj := by
     rw [K₉.symm, K₈.symm]
     exact para₂
   exact key
@@ -1050,24 +949,24 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length' (h₁ : qdr_cvx.edge_nd₁₄ ∥ q
   exact qdr_cvx.not_collinear₄₁₂
 
 /-- Given four points ABCD and Quadrilateral ABCD IsConvex, and AD ∥ BC and AD = BC, Quadrilateral ABCD is a Parallelogram_nd. -/
-theorem qdr_cvx_is_prg_nd_of_para_eq_length'_variant (h₁ : (SEG_nd A D (QDR_cvx A B C D nd cvx).nd₁₄.out) ∥ (SEG_nd B C (QDR_cvx A B C D nd cvx).nd₂₃.out)) (h₂ : (QDR_cvx A B C D nd cvx).edge_nd₁₄.length = (QDR_cvx A B C D nd cvx).edge_nd₂₃.length) : (Quadrilateral_nd.mk_is_nd nd) IsParallelogram_nd := by
+theorem qdr_cvx_is_prg_nd_of_para_eq_length'_variant (h₁ : (SEG_nd A D (QDR_cvx A B C D nd cvx).nd₁₄.out) ∥ (SEG_nd B C (QDR_cvx A B C D nd cvx).nd₂₃.out)) (h₂ : (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.length = (QDR_cvx A B C D nd cvx).edgeND₂₃.length) : (Quadrilateral_nd.mk_is_nd nd) IsParallelogram_nd := by
    unfold Quadrilateral_nd.IsParallelogram_nd
    constructor
    let permute_convex := Quadrilateral_cvx.mk_is_convex (QDR_cvx A B C D nd cvx).permute_is_convex
-   have K₁: permute_convex.edge_nd₁₂.toProj = (QDR_cvx A B C D nd cvx).edge_nd₂₃.toProj := by rfl
-   have K₂: permute_convex.edge_nd₁₂.1.length = (QDR_cvx A B C D nd cvx).edge_nd₂₃.1.length := by rfl
-   have j₂: permute_convex.edge_nd₃₄ = (QDR_cvx A B C D nd cvx).edge_nd₁₄.reverse := by rfl
-   have K₃: permute_convex.edge_nd₃₄.toProj = (QDR_cvx A B C D nd cvx).edge_nd₁₄.toProj := by
+   have K₁: permute_convex.edgeND₁₂.toProj = (QDR_cvx A B C D nd cvx).edgeND₂₃.toProj := by rfl
+   have K₂: permute_convex.edgeND₁₂.1.length = (QDR_cvx A B C D nd cvx).edgeND₂₃.1.length := by rfl
+   have j₂: permute_convex.edgeND₃₄ = (QDR_cvx A B C D nd cvx).edgeND₄₁ := by rfl
+   have K₃: permute_convex.edgeND₃₄.toProj = (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.toProj := by
      rw [j₂]
-     apply SegND.toProj_of_rev_eq_toProj
-   have K₄: permute_convex.edge_nd₃₄.1.length = (QDR_cvx A B C D nd cvx).edge_nd₁₄.1.length := by
+     apply SegND.toProj_of_rev_eq_toProj.symm
+   have K₄: permute_convex.edgeND₃₄.1.length = (QDR_cvx A B C D nd cvx).edgeND₄₁.reverse.1.length := by
      rw [j₂]
-     apply SegND.length_of_rev_eq_length
-   have H₁: permute_convex.edge_nd₁₂.toProj = permute_convex.edge_nd₃₄.toProj := by
+     apply SegND.length_of_rev_eq_length.symm
+   have H₁: permute_convex.edgeND₁₂.toProj = permute_convex.edgeND₃₄.toProj := by
      rw [K₁, K₃]
-     unfold parallel at h₁
+     unfold Parallel at h₁
      exact h₁.symm
-   have H₂: permute_convex.edge_nd₁₂.1.length = permute_convex.edge_nd₃₄.1.length := by
+   have H₂: permute_convex.edgeND₁₂.1.length = permute_convex.edgeND₃₄.1.length := by
      rw [K₂, K₄]
      exact h₂.symm
    have H: permute_convex.IsParallelogram_nd := qdr_cvx_is_prg_nd_of_para_eq_length permute_convex H₁ H₂
@@ -1076,15 +975,15 @@ theorem qdr_cvx_is_prg_nd_of_para_eq_length'_variant (h₁ : (SEG_nd A D (QDR_cv
    unfold Quadrilateral_nd.IsParallelogram_para
    unfold Quadrilateral_nd.IsParallelogram_para at para
    rcases para with ⟨para₁, para₂⟩
-   unfold parallel
-   have K₉: permute_convex.edge_nd₁₄.reverse.toProj = (QDR_cvx A B C D nd cvx).edge_nd₁₂.toProj := by rfl
-   have K₁₀: permute_convex.edge_nd₁₄.reverse.toProj = permute_convex.edge_nd₁₄.toProj := by apply SegND.toProj_of_rev_eq_toProj
-   have K₈: permute_convex.edge_nd₂₃.toProj = (QDR_cvx A B C D nd cvx).edge_nd₃₄.toProj := by rfl
+   unfold Parallel
+   have K₉: permute_convex.edgeND₄₁.toProj = (QDR_cvx A B C D nd cvx).edgeND₁₂.toProj := by rfl
+   have K₁₀: permute_convex.edgeND₄₁.toProj = permute_convex.edgeND₄₁.reverse.toProj := by apply SegND.toProj_of_rev_eq_toProj.symm
+   have K₈: permute_convex.edgeND₂₃.toProj = (QDR_cvx A B C D nd cvx).edgeND₃₄.toProj := by rfl
    rw [K₁₀] at K₉
-   unfold parallel at para₁
-   unfold parallel at para₂
+   unfold Parallel at para₁
+   unfold Parallel at para₂
    constructor
-   have key: (QDR_cvx A B C D nd cvx).edge_nd₁₂.toProj = (QDR_cvx A B C D nd cvx).edge_nd₃₄.toProj := by
+   have key: (QDR_cvx A B C D nd cvx).edgeND₁₂.toProj = (QDR_cvx A B C D nd cvx).edgeND₃₄.toProj := by
      rw [K₉.symm, K₈.symm]
      exact para₂
    exact key
@@ -1442,7 +1341,7 @@ theorem nd_eq_length_of_is_prg_nd (h : qdr.IsParallelogram_nd) : (SEG_nd qdr.poi
   rcases h with ⟨a,b⟩
   unfold Quadrilateral_nd.IsParallelogram_para at a
   rcases a with ⟨p,q⟩
-  unfold parallel at p q
+  unfold Parallel at p q
 
   rcases b with ⟨b₁,b₂,b₃,b₄⟩
   sorry
@@ -1460,7 +1359,7 @@ theorem nd_eq_length_of_is_prg_nd'  (h : qdr.IsParallelogram_nd) : (SEG_nd qdr.p
   rcases h with ⟨a,b⟩
   unfold Quadrilateral_nd.IsParallelogram_para at a
   rcases a with ⟨p,q⟩
-  unfold parallel at p q
+  unfold Parallel at p q
   rcases b with ⟨b₁,b₂,b₃,b₄⟩
   sorry
   simp only [j, dite_false] at h
diff --git a/EuclideanGeometry/Foundation/Construction/Polygon/Quadrilateral.lean b/EuclideanGeometry/Foundation/Construction/Polygon/Quadrilateral.lean
index a91c1a5e..59d29576 100644
--- a/EuclideanGeometry/Foundation/Construction/Polygon/Quadrilateral.lean
+++ b/EuclideanGeometry/Foundation/Construction/Polygon/Quadrilateral.lean
@@ -127,19 +127,43 @@ instance nd₁₄ : PtNe qdr_nd.1.4 qdr_nd.1.1 := Fact.mk qdr_nd.nd.nd₁₄
 
 /-- The edge from the first point to the second point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₁₂ : SegND P := SEG_nd qdr_nd.point₁ qdr_nd.point₂
+def edgeND₁₂ : SegND P := SEG_nd qdr_nd.point₁ qdr_nd.point₂
+
+@[simp]
+lemma edgeND₁₂_source : qdr_nd.edgeND₁₂.source = qdr_nd.point₁ := rfl
+
+@[simp]
+lemma edgeND₁₂_target : qdr_nd.edgeND₁₂.target = qdr_nd.point₂ := rfl
 
 /-- The edge from the second point to the third point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₂₃ : SegND P := SEG_nd qdr_nd.point₂ qdr_nd.point₃
+def edgeND₂₃ : SegND P := SEG_nd qdr_nd.point₂ qdr_nd.point₃
+
+@[simp]
+lemma edgeND₂₃_source : qdr_nd.edgeND₂₃.source = qdr_nd.point₂ := rfl
+
+@[simp]
+lemma edgeND₂₃_target : qdr_nd.edgeND₂₃.target = qdr_nd.point₃ := rfl
 
 /-- The edge from the third point to the fourth point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₃₄ : SegND P := SEG_nd qdr_nd.point₃ qdr_nd.point₄
+def edgeND₃₄ : SegND P := SEG_nd qdr_nd.point₃ qdr_nd.point₄
+
+@[simp]
+lemma edgeND₃₄_source : qdr_nd.edgeND₃₄.source = qdr_nd.point₃ := rfl
+
+@[simp]
+lemma edgeND₃₄_target : qdr_nd.edgeND₃₄.target = qdr_nd.point₄ := rfl
 
 /-- The edge from the fourth point to the first point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₁₄ : SegND P := SEG_nd qdr_nd.point₁ qdr_nd.point₄
+def edgeND₄₁ : SegND P := SEG_nd qdr_nd.point₄ qdr_nd.point₁
+
+@[simp]
+lemma edgeND₄₁_source : qdr_nd.edgeND₄₁.source = qdr_nd.point₄ := rfl
+
+@[simp]
+lemma edgeND₄₁_target : qdr_nd.edgeND₄₁.target = qdr_nd.point₁ := rfl
 
 /--angle at point₁ of qdr_nd-/
 @[pp_dot]
@@ -253,15 +277,15 @@ variable {P : Type _} [EuclideanPlane P] {A B C D : P}
 
 structure is_convex_of_three_sides_of_same_side' where
   qdr_nd : Quadrilateral_nd P
-  same_side₁ : odist_sign qdr_nd.point₁ qdr_nd.edge_nd₃₄ = odist_sign qdr_nd.point₂ qdr_nd.edge_nd₃₄
-  same_side₂ : odist_sign qdr_nd.point₂ qdr_nd.edge_nd₁₄.reverse = odist_sign qdr_nd.point₃ qdr_nd.edge_nd₁₄.reverse
-  same_side₃ : odist_sign qdr_nd.point₃ qdr_nd.edge_nd₁₂ = odist_sign qdr_nd.point₄ qdr_nd.edge_nd₁₂
+  same_side₁ : odist_sign qdr_nd.point₁ qdr_nd.edgeND₃₄ = odist_sign qdr_nd.point₂ qdr_nd.edgeND₃₄
+  same_side₂ : odist_sign qdr_nd.point₂ qdr_nd.edgeND₄₁ = odist_sign qdr_nd.point₃ qdr_nd.edgeND₄₁
+  same_side₃ : odist_sign qdr_nd.point₃ qdr_nd.edgeND₁₂ = odist_sign qdr_nd.point₄ qdr_nd.edgeND₁₂
 
 /- Given Quadrilateral_nd qdr_nd, if qdr_nd.point₁ and qdr_nd.point₂ are at the same side of qdr_nd.nd₃₄, and it also holds for nd₄₁ and nd₁₂, then it's convex. -/
 theorem is_convex_of_three_sides_of_same_side (p : is_convex_of_three_sides_of_same_side' (P := P)) : p.qdr_nd IsConvex := by
   let qdr_nd := p.qdr_nd
   sorry
-  -- by_cases h : odist_sign qdr_nd.point₁ qdr_nd.edge_nd₃₄ = 1
+  -- by_cases h : odist_sign qdr_nd.point₁ qdr_nd.edgeND₃₄ = 1
   --
 
 structure is_convex_of_diag_inx_lies_int' where
@@ -466,50 +490,50 @@ instance nd₁₄ : PtNe qdr_cvx.point₄ qdr_cvx.point₁ := by infer_instance
 
 /-- The edge from the first point to the second point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₁₂ : SegND P := SEG_nd qdr_cvx.point₁ qdr_cvx.point₂
+def edgeND₁₂ : SegND P := SEG_nd qdr_cvx.point₁ qdr_cvx.point₂
 
 /-- The edge from the second point to the third point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₂₃ : SegND P := SEG_nd qdr_cvx.point₂ qdr_cvx.point₃
+def edgeND₂₃ : SegND P := SEG_nd qdr_cvx.point₂ qdr_cvx.point₃
 
 /-- The edge from the third point to the fourth point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₃₄ : SegND P := SEG_nd qdr_cvx.point₃ qdr_cvx.point₄
+def edgeND₃₄ : SegND P := SEG_nd qdr_cvx.point₃ qdr_cvx.point₄
 
 /-- The edge from the fourth point to the first point of a quadrilateral -/
 @[pp_dot]
-def edge_nd₁₄ : SegND P := SEG_nd qdr_cvx.point₁ qdr_cvx.point₄
+def edgeND₄₁.reverse : SegND P := SEG_nd qdr_cvx.point₁ qdr_cvx.point₄
 -/
 
 /-- Given a convex quadrilateral qdr_cvx, its 1st, 2nd and 3rd points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₁ point₂}$ is not the same as the projective direction of the vector $\overrightarrow{point₁ point₃}$. -/
-theorem not_collinear₁₂₃ : ¬ collinear qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₃ := sorry
+theorem not_collinear₁₂₃ : ¬ Collinear qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₃ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 2nd, 3rd and 4th points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₂ point₃}$ is not the same as the projective direction of the vector $\overrightarrow{point₂ point₄}$. -/
-theorem not_collinear₂₃₄ : ¬ collinear qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.point₄ := sorry
+theorem not_collinear₂₃₄ : ¬ Collinear qdr_cvx.point₂ qdr_cvx.point₃ qdr_cvx.point₄ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 3rd, 4th and 1st points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₃ point₄}$ is not the same as the projective direction of the vector $\overrightarrow{point₃ point₁}$. -/
-theorem not_collinear₃₄₁ : ¬ collinear qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ := sorry
+theorem not_collinear₃₄₁ : ¬ Collinear qdr_cvx.point₃ qdr_cvx.point₄ qdr_cvx.point₁ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 4th, 1st and 3rd points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₃ point₄}$ is not the same as the projective direction of the vector $\overrightarrow{point₃ point₁}$. -/
-theorem not_collinear₄₁₃ : ¬ collinear qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₃ := sorry
+theorem not_collinear₄₁₃ : ¬ Collinear qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₃ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 4th, 3rd and 1st points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₃ point₄}$ is not the same as the projective direction of the vector $\overrightarrow{point₃ point₁}$. -/
-theorem not_collinear₄₃₁ : ¬ collinear qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ := sorry
+theorem not_collinear₄₃₁ : ¬ Collinear qdr_cvx.point₄ qdr_cvx.point₃ qdr_cvx.point₁ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 1st, 4th and 3rd points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₃ point₄}$ is not the same as the projective direction of the vector $\overrightarrow{point₃ point₁}$. -/
-theorem not_collinear₁₄₃ : ¬ collinear qdr_cvx.point₁ qdr_cvx.point₄ qdr_cvx.point₃ := sorry
+theorem not_collinear₁₄₃ : ¬ Collinear qdr_cvx.point₁ qdr_cvx.point₄ qdr_cvx.point₃ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 4th, 1st and 2nd points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₄ point₁}$ is not the same as the projective direction of the vector $\overrightarrow{point₄ point₂}$. -/
-theorem not_collinear₄₁₂ : ¬ collinear qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₂ := sorry
+theorem not_collinear₄₁₂ : ¬ Collinear qdr_cvx.point₄ qdr_cvx.point₁ qdr_cvx.point₂ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 2nd, 1st and 4th points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₄ point₁}$ is not the same as the projective direction of the vector $\overrightarrow{point₄ point₂}$. -/
-theorem not_collinear₂₁₄ : ¬ collinear qdr_cvx.point₂ qdr_cvx.point₁ qdr_cvx.point₄ := sorry
+theorem not_collinear₂₁₄ : ¬ Collinear qdr_cvx.point₂ qdr_cvx.point₁ qdr_cvx.point₄ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 1st, 4th and 2nd points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₄ point₁}$ is not the same as the projective direction of the vector $\overrightarrow{point₄ point₂}$. -/
-theorem not_collinear₁₄₂ : ¬ collinear qdr_cvx.point₁ qdr_cvx.point₄ qdr_cvx.point₂ := sorry
+theorem not_collinear₁₄₂ : ¬ Collinear qdr_cvx.point₁ qdr_cvx.point₄ qdr_cvx.point₂ := sorry
 
 /-- Given a convex quadrilateral qdr_cvx, its 1st, 2nd and 4th points are not collinear, i.e. the projective direction of the vector $\overrightarrow{point₄ point₁}$ is not the same as the projective direction of the vector $\overrightarrow{point₄ point₂}$. -/
-theorem not_collinear₁₂₄ : ¬ collinear qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ := sorry
+theorem not_collinear₁₂₄ : ¬ Collinear qdr_cvx.point₁ qdr_cvx.point₂ qdr_cvx.point₄ := sorry
 
 -- We need to add a bunch of such theorems as they may be useful in discussing general quadrilaterals, i.e. not convex, even as contradictory in proofs.
 
diff --git a/EuclideanGeometry/Foundation/Construction/Triangle/AngleBisector.lean b/EuclideanGeometry/Foundation/Construction/Triangle/AngleBisector.lean
index 4e0ea66f..f4a6d9d1 100644
--- a/EuclideanGeometry/Foundation/Construction/Triangle/AngleBisector.lean
+++ b/EuclideanGeometry/Foundation/Construction/Triangle/AngleBisector.lean
@@ -177,12 +177,12 @@ theorem angbisline_of_angle₁_angle₂_not_parallel {tri_nd : TriangleND P} : 
     rw [← h₂] at g
     sorry
   have er : A₁.end_ray.toDirLine = A₂.end_ray.toDirLine.reverse := by
-    have h₃ : A₁.end_ray = tri_nd.edge_nd₃.toRay := rfl
-    have h₄ : A₂.end_ray = tri_nd.edge_nd₃.reverse.toRay := rfl
+    have h₃ : A₁.end_ray = tri_nd.edgeND₃.toRay := rfl
+    have h₄ : A₂.end_ray = tri_nd.edgeND₃.reverse.toRay := rfl
     rw [h₃]
     rw [h₄]
-    have h₅ : tri_nd.edge_nd₃.reverse.toDirLine.reverse = tri_nd.edge_nd₃.reverse.reverse.toDirLine := by rw [SegND.toDirLine_rev_eq_rev_toDirLine]
-    have h₆ : tri_nd.edge_nd₃.reverse.reverse.toDirLine = tri_nd.edge_nd₃.toDirLine := rfl
+    have h₅ : tri_nd.edgeND₃.reverse.toDirLine.reverse = tri_nd.edgeND₃.reverse.reverse.toDirLine := by rw [SegND.toDirLine_rev_eq_rev_toDirLine]
+    have h₆ : tri_nd.edgeND₃.reverse.reverse.toDirLine = tri_nd.edgeND₃.toDirLine := rfl
     rw [h₆] at h₅
     exact h₅.symm
   have g₁ : IsConsecutiveIntAng A₁ A₂ := ⟨sr, er⟩
diff --git a/EuclideanGeometry/Foundation/Tactic/Collinear/perm_collinear.lean b/EuclideanGeometry/Foundation/Tactic/Collinear/perm_collinear.lean
index a8b5295e..4f2cedaa 100644
--- a/EuclideanGeometry/Foundation/Tactic/Collinear/perm_collinear.lean
+++ b/EuclideanGeometry/Foundation/Tactic/Collinear/perm_collinear.lean
@@ -8,7 +8,7 @@ variable {P : Type _} [EuclideanPlane P]
 
 def extractCollinear (expr : Q(Prop)) : MetaM (Option Expr) :=
   match expr with
-  | ~q(@EuclidGeom.collinear _ (_) _ _ _) =>
+  | ~q(@EuclidGeom.Collinear _ (_) _ _ _) =>
       return .some expr
   | _ => return .none
 
@@ -35,31 +35,31 @@ def evalPerm_collinear : Tactic := fun stx =>
       catch
         _ => pure ()
       try
-        let t <- `(tactic| refine flip_collinear_fst_snd $x0)
+        let t <- `(tactic| refine Collinear.perm₁₃₂ $x0)
         evalTactic t
         return
       catch
         _ => pure ()
       try
-        let t <- `(tactic| refine flip_collinear_snd_trd $x0)
+        let t <- `(tactic| refine Collinear.perm₂₁₃ $x0)
         evalTactic t
         return
       catch
         _ => pure ()
       try
-        let t <- `(tactic| refine flip_collinear_fst_snd (flip_collinear_snd_trd $x0))
+        let t <- `(tactic| refine Collinear.perm₃₁₂ $x0)
         evalTactic t
         return
       catch
         _ => pure ()
       try
-        let t <- `(tactic| refine flip_collinear_snd_trd (flip_collinear_fst_snd $x0))
+        let t <- `(tactic| refine Collinear.perm₂₃₁ $x0)
         evalTactic t
         return
       catch
         _ => pure ()
       try
-        let t <- `(tactic| refine flip_collinear_fst_snd (flip_collinear_snd_trd (flip_collinear_fst_snd $x0)))
+        let t <- `(tactic| refine Collinear.perm₃₂₁ $x0)
         evalTactic t
         return
       catch
@@ -68,7 +68,7 @@ def evalPerm_collinear : Tactic := fun stx =>
   | _ => throwUnsupportedSyntax
 
 
-example {A B C : P} {h : collinear B C A} : collinear A B C := by
+example {A B C : P} {h : Collinear B C A} : Collinear A B C := by
   perm_collinear
 
 end EuclidGeom
diff --git a/EuclideanGeometry/Unused/Congruence.lean b/EuclideanGeometry/Unused/Congruence.lean
index dc642119..af52f3ca 100644
--- a/EuclideanGeometry/Unused/Congruence.lean
+++ b/EuclideanGeometry/Unused/Congruence.lean
@@ -492,9 +492,9 @@ theorem cosine_eq_of_SSS (e₁ : tr_nd₁.1.edge₁.length = tr_nd₂.1.edge₁.
   rw [e₁,e₂,e₃,←cos₂] at cos₁
   field_simp at cos₁
   have u₁ : 0 < tr_nd₂.1.edge₃.length := by
-    exact length_pos_iff_nd.mpr tr_nd₂.edge_nd₃.2
+    exact length_pos_iff_nd.mpr tr_nd₂.edgeND₃.2
   have u₂ : 0 < tr_nd₂.1.edge₂.length := by
-    exact length_pos_iff_nd.mpr tr_nd₂.edge_nd₂.2
+    exact length_pos_iff_nd.mpr tr_nd₂.edgeND₂.2
   have h0 : (tr_nd₂.1.edge₃.length * tr_nd₂.1.edge₂.length) > 0 := by
     field_simp [u₁,u₂]
   rcases cos₁ with x | y
diff --git a/EuclideanGeometry/Unused/Line.lean b/EuclideanGeometry/Unused/Line.lean
index 6b1b7bf9..186216a7 100644
--- a/EuclideanGeometry/Unused/Line.lean
+++ b/EuclideanGeometry/Unused/Line.lean
@@ -4,17 +4,17 @@ noncomputable section
 namespace EuclidGeom
 
 @[ext]
-class Line (P : Type _) [EuclideanPlane P] where 
+class Line (P : Type _) [EuclideanPlane P] where
   carrier : Set P
-  linear : ∀ (A B C : P), (A ∈ carrier) → (B ∈ carrier) → (C ∈ carrier) → collinear A B C
-  maximal : ∀ (A B : P), (A ∈ carrier) → (B ∈ carrier) → (B ≠ A) → (∀ (C : P), collinear A B C → (C ∈ carrier))
+  linear : ∀ (A B C : P), (A ∈ carrier) → (B ∈ carrier) → (C ∈ carrier) → Collinear A B C
+  maximal : ∀ (A B : P), (A ∈ carrier) → (B ∈ carrier) → (B ≠ A) → (∀ (C : P), Collinear A B C → (C ∈ carrier))
   nontriv : ∃ (A B : P), (A ∈ carrier) ∧ (B ∈ carrier) ∧ (B ≠ A)
 
 namespace Line
 
-variable  {P : Type _} [EuclideanPlane P] 
+variable  {P : Type _} [EuclideanPlane P]
 
--- define a line from two points 
+-- define a line from two points
 
 def mk_pt_pt (A B : P) (h : B ≠ A) : Line P where
   carrier := {C : P | ∃ t : ℝ, VEC A C = t • VEC A B}
@@ -36,13 +36,13 @@ def mk_pt_pt (A B : P) (h : B ≠ A) : Line P where
     · have h' : (ty = tx) ∨ (tz = tx) ∨ (ty = tz) := by tauto
       by_cases ty = tx
       · rw [pt_eq_pt_of_eq_smul_smul h hy hx]
-        exact triv_collinear _ _
+        exact triv_collinear₁₂ _ _
       · by_cases tz = tx
         · rw [pt_eq_pt_of_eq_smul_smul h hz hx]
-          exact flip_collinear_snd_trd $ triv_collinear _ _
+          exact Collinear.perm₁₃₂ $ triv_collinear₁₂ _ _
         · have h : ty = tz := by tauto
           rw [pt_eq_pt_of_eq_smul_smul h hy hz]
-          exact flip_collinear_fst_snd $ flip_collinear_snd_trd $ triv_collinear _ _
+          exact Collinear.perm₂₁₃ $ Collinear.perm₁₃₂ $ triv_collinear₁₂ _ _
   maximal x y := by
     unfold Membership.mem Set.instMembershipSet Set.Mem setOf
     simp only [forall_exists_index]
@@ -67,7 +67,7 @@ def mk_pt_pt (A B : P) (h : B ≠ A) : Line P where
 
 end Line
 
-scoped notation "LIN" => Line.mk_pt_pt 
+scoped notation "LIN" => Line.mk_pt_pt
 
 namespace Line
 
@@ -82,12 +82,12 @@ instance : Carrier P (Line P) where
 
 end Line
 
--- Now we introduce useful theorems to avoid using more unfolds in further proofs. 
+-- Now we introduce useful theorems to avoid using more unfolds in further proofs.
 variable {P : Type _} [EuclideanPlane P]
 
 section Compaitiblity_of_coercions_of_mk_pt_pt
 
--- The first point and the second point in Line.mk_pt_pt LiesOn the line it make. 
+-- The first point and the second point in Line.mk_pt_pt LiesOn the line it make.
 
 theorem fst_pt_lies_on_line_of_pt_pt {A B : P} (h : B ≠ A) : A LiesOn LIN A B h := by
   unfold lies_on Carrier.carrier Line.instCarrierLine
@@ -110,7 +110,7 @@ theorem pt_lies_on_line_of_pt_pt_of_ne {A B : P} (h: B ≠ A) : A LiesOn LIN A B
   exact fst_pt_lies_on_line_of_pt_pt h
   exact snd_pt_lies_on_line_of_pt_pt h
 
-theorem lies_on_line_of_pt_pt_iff_collinear {A B : P} (h : B ≠ A) : ∀ X : P, (X LiesOn (LIN A B h)) ↔ collinear A B X := by
+theorem lies_on_line_of_pt_pt_iff_collinear {A B : P} (h : B ≠ A) : ∀ X : P, (X LiesOn (LIN A B h)) ↔ Collinear A B X := by
   intro X
   constructor
   intro hx
@@ -164,7 +164,7 @@ def Line.toProj (l : Line P) : Proj :=
   -- by choose pr _ using (exist_unique_proj_of_line l)
   -- exact pr
 
--- If you don't want to use Classical.choose, please use this theorem to simplify your Line.toProj. 
+-- If you don't want to use Classical.choose, please use this theorem to simplify your Line.toProj.
 
 theorem line_toProj_eq_seg_nd_toProj_of_lies_on {A B : P} {l : Line P} (ha : A LiesOn l) (hb : B LiesOn l) (hab : B ≠ A) : SegND.toProj ⟨SEG A B, hab⟩ = l.toProj := (Classical.choose_spec (exist_unique_proj_of_line l)).1 A B ha hb hab
 
@@ -180,7 +180,7 @@ section Compatibility_of_LiesOn
 
 -- This is also a typical proof that shows how to use the four conditions in the def of a line. Please write it shorter in future.
 
-theorem lies_on_iff_collinear_of_ne_lies_on_lies_on {A B : P} {l : Line P} (h : B ≠ A) (ha : A LiesOn l) (hb : B LiesOn l) : ∀ C : P, (C LiesOn l) ↔ collinear A B C := by
+theorem lies_on_iff_collinear_of_ne_lies_on_lies_on {A B : P} {l : Line P} (h : B ≠ A) (ha : A LiesOn l) (hb : B LiesOn l) : ∀ C : P, (C LiesOn l) ↔ Collinear A B C := by
   intro C
   constructor
   intro hc
@@ -219,13 +219,13 @@ theorem eq_line_of_pt_pt_of_ne {A B : P} {l : Line P} (h : B ≠ A) (ha : A Lies
 
 theorem eq_of_pt_pt_lies_on_of_ne {A B : P} (h : B ≠ A) {l₁ l₂ : Line P}(hA₁ : A LiesOn l₁) (hB₁ : B LiesOn l₁) (hA₂ : A LiesOn l₂) (hB₂ : B LiesOn l₂) : l₁ = l₂ := sorry
 
-theorem collinear_iff_exist_line_lies_on (A B C : P) : collinear A B C ↔ ∃ l : Line P, (A LiesOn l) ∧ (B LiesOn l) ∧ (C LiesOn l) := by
+theorem collinear_iff_exist_line_lies_on (A B C : P) : Collinear A B C ↔ ∃ l : Line P, (A LiesOn l) ∧ (B LiesOn l) ∧ (C LiesOn l) := by
   sorry
 
 end Compatibility_of_LiesOn
 /- def coe from ray to line-/
 
-def Ray.toLine (r : Ray P) := LIN r.source (r.toDir.toVec +ᵥ r.source) (by 
+def Ray.toLine (r : Ray P) := LIN r.source (r.toDir.toVec +ᵥ r.source) (by
   simp only [ne_eq, vadd_eq_self_iff_vec_eq_zero]
   exact Dir.toVec_ne_zero r.toDir)
 
@@ -284,12 +284,12 @@ theorem lies_on_of_SegND_toProj_eq_toProj {A B : P} {l : Line P} (ha : A LiesOn
   let g := line_toProj_eq_seg_nd_toProj_of_lies_on ha h.1 h.2
   rw [← hp] at g
   unfold SegND.toProj SegND.toVecND at g
-  simp only [ne_eq] at g 
-  have c : collinear A X B := by
-    rw [← iff_true (collinear A X B), ← eq_iff_iff]
-    unfold collinear collinear_of_nd
+  simp only [ne_eq] at g
+  have c : Collinear A X B := by
+    rw [← iff_true (Collinear A X B), ← eq_iff_iff]
+    unfold Collinear collinear_of_nd
     simp [g]
-    by_cases (B = X ∨ A = B ∨ X = A) 
+    by_cases (B = X ∨ A = B ∨ X = A)
     · simp only [h, dite_eq_ite]
     · simp only [h, dite_eq_ite]
   exact (lies_on_iff_collinear_of_ne_lies_on_lies_on h.2 ha h.1 B).2 c
@@ -307,8 +307,8 @@ end Archimedean_property
 section Line_passing_point_with_given_Proj
 
 theorem exist_line_of_pt_proj (A : P) (pr : Proj) : ∃ l : Line P, A LiesOn l ∧ l.toProj = pr := by
-  rcases Quot.exists_rep pr with ⟨dir, hd⟩ 
-  let r : Ray P := ⟨A, dir⟩ 
+  rcases Quot.exists_rep pr with ⟨dir, hd⟩
+  let r : Ray P := ⟨A, dir⟩
   use r.toLine
   constructor
   exact Ray.lies_on_toLine_of_lie_on (Ray.source_lies_on r)
@@ -326,12 +326,12 @@ theorem exist_unique_line_of_pt_proj (A : P) (pr : Proj) : ∃! l : Line P, A Li
     simp only [ne_eq, vadd_eq_self_iff_vec_eq_zero, Dir.toVec_ne_zero dir, not_false_eq_true]
   apply (lies_on_iff_lies_on_iff_line_eq_line l₂ l₁).1
   intro X
-  by_cases X = A 
+  by_cases X = A
   · rw [h]
     tauto
   · rw [lies_on_iff_eq_toProj_of_lies_on hl₁.1 h, hl₁.2, lies_on_iff_eq_toProj_of_lies_on hl₂.1 h, hl₂.2]
 
-def Line.mk_pt_proj (A : P) (pr : Proj) : Line P := 
+def Line.mk_pt_proj (A : P) (pr : Proj) : Line P :=
   Classical.choose (exist_unique_line_of_pt_proj A pr)
 
 theorem pt_lies_on_and_proj_eq_of_line_mk_pt_proj (A : P) (pr : Proj) : A LiesOn (Line.mk_pt_proj A pr) ∧ (Line.mk_pt_proj A pr).toProj = pr := by
@@ -339,4 +339,4 @@ theorem pt_lies_on_and_proj_eq_of_line_mk_pt_proj (A : P) (pr : Proj) : A LiesOn
 
 end Line_passing_point_with_given_Proj
 
-end EuclidGeom
\ No newline at end of file
+end EuclidGeom
diff --git a/EuclideanGeometry/Unused/Parallel.lean b/EuclideanGeometry/Unused/Parallel.lean
index 9890eacd..70a11350 100644
--- a/EuclideanGeometry/Unused/Parallel.lean
+++ b/EuclideanGeometry/Unused/Parallel.lean
@@ -61,7 +61,7 @@ scoped infix : 50 " LiesOnarObj " => LinearObj.IsOnLinearObj
 
 -- Our definition of parallel for LinearObj is very general. Not only can it apply to different types of Objs, but also include degenerate cases, such as ⊆(inclusions), =(equal).
 
-def parallel' {α β: Type _} (l₁ : α) (l₂ : β) [Coe α (LinearObj P)] [Coe β (LinearObj P)] : Prop :=  LinearObj.toProj (P := P) (Coe.coe l₁) = LinearObj.toProj (P := P) (Coe.coe l₂)
+def Parallel' {α β: Type _} (l₁ : α) (l₂ : β) [Coe α (LinearObj P)] [Coe β (LinearObj P)] : Prop :=  LinearObj.toProj (P := P) (Coe.coe l₁) = LinearObj.toProj (P := P) (Coe.coe l₂)
 
 -- class PlaneFigure' (P : Type _) [EuclideanPlane P] {α : Type _} where
 
@@ -69,22 +69,22 @@ def parallel' {α β: Type _} (l₁ : α) (l₂ : β) [Coe α (LinearObj P)] [Co
 
 
 
-def parallel (l₁ l₂: LinearObj P) : Prop := l₁.toProj = l₂.toProj
+def Parallel (l₁ l₂: LinearObj P) : Prop := l₁.toProj = l₂.toProj
 
-instance : IsEquiv (LinearObj P) parallel where
+instance : IsEquiv (LinearObj P) Parallel where
   refl _ := rfl
   symm _ _ := Eq.symm
   trans _ _ _ := Eq.trans
 
-scoped infix : 50 " ParallelTo " => parallel
+scoped infix : 50 " ParallelTo " => Parallel
 
-scoped infix : 50 " ∥ " => parallel
+scoped infix : 50 " ∥ " => Parallel
 
 /- lots of trivial parallel relation of vec of 2 pt lies on Line, coercions, ... -/
 
 section parallel_theorem
 
-theorem ray_parallel_toLine_assoc_ray (ray : Ray P) :  parallel (LinearObj.ray ray) ray.toLine := sorry
+theorem ray_parallel_toLine_assoc_ray (ray : Ray P) :  Parallel (LinearObj.ray ray) ray.toLine := sorry
 
 theorem seg_parallel_toRay_assoc_seg_of_nontriv (seg_nd : SegND P) : LinearObj.seg_nd seg_nd ∥ seg_nd.toRay := sorry
 
diff --git a/EuclideanGeometry/Unused/Parallelogram.lean b/EuclideanGeometry/Unused/Parallelogram.lean
index cc685ba5..319900a3 100644
--- a/EuclideanGeometry/Unused/Parallelogram.lean
+++ b/EuclideanGeometry/Unused/Parallelogram.lean
@@ -7,8 +7,8 @@ noncomputable section
 namespace EuclidGeom
 
 class Parallelogram (P : Type _) [EuclideanPlane P] extends Quadrilateral_cvx P where
-  para : LinearObj.seg_nd toQuadrilateral_cvx.edge_nd₁₂ ∥ toQuadrilateral_cvx.edge_nd₃₄
-  para' : LinearObj.seg_nd toQuadrilateral_cvx.edge_nd₂₃ ∥ toQuadrilateral_cvx.edge_nd₄₁
+  para : LinearObj.seg_nd toQuadrilateral_cvx.edgeND₁₂ ∥ toQuadrilateral_cvx.edgeND₃₄
+  para' : LinearObj.seg_nd toQuadrilateral_cvx.edgeND₂₃ ∥ toQuadrilateral_cvx.edgeND₄₁
 
 /-
 section make
@@ -23,7 +23,7 @@ end Parallelogram
 end make
 -/
 
-def Quadrilateral_cvx.is_parallelogram {P : Type _} [EuclideanPlane P] (qdr_cvx : Quadrilateral_cvx P) : Prop := (LinearObj.seg_nd qdr_cvx.edge_nd₁₂ ∥ qdr_cvx.edge_nd₃₄) ∧ (LinearObj.seg_nd qdr_cvx.edge_nd₂₃ ∥ qdr_cvx.edge_nd₄₁)
+def Quadrilateral_cvx.is_parallelogram {P : Type _} [EuclideanPlane P] (qdr_cvx : Quadrilateral_cvx P) : Prop := (LinearObj.seg_nd qdr_cvx.edgeND₁₂ ∥ qdr_cvx.edgeND₃₄) ∧ (LinearObj.seg_nd qdr_cvx.edgeND₂₃ ∥ qdr_cvx.edgeND₄₁)
 
 def Quadrilateral.is_parallelogram {P : Type _} [EuclideanPlane P] (qdr : Quadrilateral P) : Prop := by
   by_cases qdr.IsConvex 
diff --git a/EuclideanGeometry/Unused/Unused.lean b/EuclideanGeometry/Unused/Unused.lean
index b375008c..97a83b27 100644
--- a/EuclideanGeometry/Unused/Unused.lean
+++ b/EuclideanGeometry/Unused/Unused.lean
@@ -250,7 +250,7 @@ class Triangle' (P : Type _) [EuclideanPlane P] where
   point₁ : P
   point₂ : P
   point₃ : P
-  nontriv : ¬ collinear point₁ point₂ point₃
+  nontriv : ¬ Collinear point₁ point₂ point₃
 
 namespace Triangle
 
@@ -286,20 +286,20 @@ scoped infix : 50 " IsInsideLTriangle " => Triangle.IsInside
 
 end nondeg_tri
 
-section collinear
+section Collinear
 
 variable {P : Type _} [EuclideanPlane P]
 
-theorem collinear_ACB_of_collinear_ABC {A B C : P} (h : collinear A B C): collinear A C B := sorry
+theorem collinear_ACB_of_collinear_ABC {A B C : P} (h : Collinear A B C): Collinear A C B := sorry
 
-theorem collinear_BAC_of_collinear_ABC {A B C : P} (h : collinear A B C): collinear B A C := sorry
+theorem collinear_BAC_of_collinear_ABC {A B C : P} (h : Collinear A B C): Collinear B A C := sorry
 
-theorem collinear_BCA_of_collinear_ABC {A B C : P} (h : collinear A B C): collinear B C A := sorry
+theorem collinear_BCA_of_collinear_ABC {A B C : P} (h : Collinear A B C): Collinear B C A := sorry
 
-theorem collinear_CAB_of_collinear_ABC {A B C : P} (h : collinear A B C): collinear C A B := sorry
+theorem collinear_CAB_of_collinear_ABC {A B C : P} (h : Collinear A B C): Collinear C A B := sorry
 
-theorem collinear_CBA_of_collinear_ABC {A B C : P} (h : collinear A B C): collinear C B A := sorry
-end collinear
+theorem collinear_CBA_of_collinear_ABC {A B C : P} (h : Collinear A B C): Collinear C B A := sorry
+end Collinear
 
 /-!
 section HasFallsOn
@@ -355,12 +355,12 @@ scoped notation p "IsIntersectionOf" A B => IsIntersectionPoint p A B
 class HasProj (α : Type _) where
   toProj : (α → Proj)
 
-def parallel {α β : Type _} (A : α) (B : β) [HasProj α] [HasProj β] : Prop := HasProj.toProj A = HasProj.toProj B
+def Parallel {α β : Type _} (A : α) (B : β) [HasProj α] [HasProj β] : Prop := HasProj.toProj A = HasProj.toProj B
 
-scoped notation A "IsParallelTo" B => parallel A B
-scoped notation A "∥" B => parallel A B
+scoped notation A "IsParallelTo" B => Parallel A B
+scoped notation A "∥" B => Parallel A B
 
-namespace parallel
+namespace Parallel
 
 protected theorem refl {α : Type _} (A : α) [HasProj α] : A ∥ A := rfl
 
@@ -368,20 +368,20 @@ protected theorem symm {α β : Type _} (A : α) (B : β) [HasProj α] [HasProj
 
 protected theorem trans {α β γ : Type _} (A : α) (B : β) (C : γ) [HasProj α] [HasProj β] [HasProj γ]: (A ∥ B) → (B ∥ C) → (A ∥ C) := Eq.trans
 
-end parallel
+end Parallel
 
-def perpendicular {α β : Type _} (A : α) (B : β) [HasProj α] [HasProj β] : Prop := sorry
+def Perpendicular {α β : Type _} (A : α) (B : β) [HasProj α] [HasProj β] : Prop := sorry
 
-scoped notation A "IsPerpendicularTo" B => perpendicular A B
-scoped notation A "⟂" B => perpendicular A B
+scoped notation A "IsPerpendicularTo" B => Perpendicular A B
+scoped notation A "⟂" B => Perpendicular A B
 
-namespace perpendicular
+namespace Perpendicular
 
 protected theorem irrefl {α : Type _} (A : α) [HasProj α] : ¬ (A ⟂ A) := by sorry
 
 protected theorem symm {α β : Type _} (A : α) (B : β) [HasProj α] [HasProj β] : (A ⟂ B) → (B ⟂ A) := sorry
 
-end perpendicular
+end Perpendicular
 
 theorem parallel_of_perp_perp {α β γ : Type _} (A : α) (B : β) (C : γ) [HasProj α] [HasProj β] [HasProj γ] : (A ⟂ B) → (B ⟂ C) → (A ∥ C)  := sorry
 -/ -/
diff --git a/EuclideanGeometry/Unused/notes-2023-20-19.txt b/EuclideanGeometry/Unused/notes-2023-20-19.txt
index c84d5332..41a44b34 100644
--- a/EuclideanGeometry/Unused/notes-2023-20-19.txt
+++ b/EuclideanGeometry/Unused/notes-2023-20-19.txt
@@ -7,7 +7,7 @@ Structure of Tactic: LinearOrder
 
 Input: 
   ⬝ goal (check that it is of the form C LiesOn/LiesIn SEG_nd A B h, or extensions)
-  ⬝ get all known hypotheses including, LiesOn/LiesIn, extension, collinear, ¬ ▵.IsND, points being equal, and etc. (Each such statement may also provide certain information regarding point positions)
+  ⬝ get all known hypotheses including, LiesOn/LiesIn, extension, Collinear, ¬ ▵.IsND, points being equal, and etc. (Each such statement may also provide certain information regarding point positions)
 
 Additional concepts/theorems to be established:
   ⬝ Given  RAY A B h, and two points C D such that C D LiesOn LIN A B h,  whether C < D for the direction or now, and related compatibility theorems, and conversion theorems to LiesOn/LiesIn etc.
diff --git a/Plan_files/DetailedPlan/Axiom-Linear.tex b/Plan_files/DetailedPlan/Axiom-Linear.tex
index 569376ca..917d0d28 100644
--- a/Plan_files/DetailedPlan/Axiom-Linear.tex
+++ b/Plan_files/DetailedPlan/Axiom-Linear.tex
@@ -149,7 +149,7 @@ \section{Content in Colinarity.lean}
 
   Definitions :
     (defn) collinear_of_nd : Prop -- Given three distinct points (A B C : P), return whether the three points are collinear, i.e. whether (VEC A B).toProj = (VEC A C).toProj
-    (defn) collinear : Prop -- Given three points (A B C : P), return whether they are collinear; if at least two of them are equal, then they are considered automatically collinear.
+    (defn) Collinear : Prop -- Given three points (A B C : P), return whether they are collinear; if at least two of them are equal, then they are considered automatically collinear.
   
   Theorems :
     collinear_of_vec_eq_smul_vec -- Given three points (A B C : P), if VEC A C = t ⬝ VEC A B for some t ∈ ℝ, then A B C are collinear
diff --git a/Plan_files/DetailedPlan/Axiom-Linear.txt b/Plan_files/DetailedPlan/Axiom-Linear.txt
index 6a3481e7..c3c1e322 100644
--- a/Plan_files/DetailedPlan/Axiom-Linear.txt
+++ b/Plan_files/DetailedPlan/Axiom-Linear.txt
@@ -363,13 +363,13 @@ Theorem \verb|collinear_of_vec_eq_smul_vec'|: Given three points $A$, $B$, $C$,
 Theorem \verb|collinear_iff_eq_smul_vec_of_ne|: Given three points $A$, $B$, $C$ such that $B \neq A$, we have $A$, $B$, $C$ are collinear if and only if the vector $\overrightarrow{AC}$ is a scalar multiple of the vector $\overrightarrow{AB}$.
 
 \item 
-Theorem \verb|triv_collinear|: For any two points $A$ and $C$, the points $A$, $A$, $C$ are collinear.
+Theorem \verb|triv_collinear₁₂|: For any two points $A$ and $C$, the points $A$, $A$, $C$ are collinear.
 
 \item 
-Theorem \verb|flip_collinear_snd_trd|: Given three points $A$, $B$, and $C$, if $A$, $B$, $C$ are collinear (in that order), then $A$, $C$, $B$ are collinear (in that order); in other words, swapping the last two of the three points does not change the definition of colinarity.
+Theorem \verb|Collinear.perm₁₃₂|: Given three points $A$, $B$, and $C$, if $A$, $B$, $C$ are collinear (in that order), then $A$, $C$, $B$ are collinear (in that order); in other words, swapping the last two of the three points does not change the definition of colinarity.
 
 \item 
-Theorem \verb|flip_collinear_fst_snd|: Given three points $A$, $B$, and $C$, if $A$, $B$, $C$ are collinear (in that order), then $B$, $A$, $C$ are collinear (in that order); in other words, in the definition of colinarity, swapping the first two of the three points does not change property of the three points being collinear.
+Theorem \verb|Collinear.perm₂₁₃|: Given three points $A$, $B$, and $C$, if $A$, $B$, $C$ are collinear (in that order), then $B$, $A$, $C$ are collinear (in that order); in other words, in the definition of colinarity, swapping the first two of the three points does not change property of the three points being collinear.
 
 \item 
 Theorem \verb|collinear_of_collinear_collinear_ne|: Given four points $A$, $B$, $C$, $D$ with $B \neq A$, if $A$, $B$, $C$ are collinear, and if $A$, $B$, $D$ are collinear, then $A$, $C$, $D$ are collinear.
diff --git a/Plan_files/DetailedPlan/Axiom-Linear_Object.lean b/Plan_files/DetailedPlan/Axiom-Linear_Object.lean
index 059eb579..6a68c0af 100644
--- a/Plan_files/DetailedPlan/Axiom-Linear_Object.lean
+++ b/Plan_files/DetailedPlan/Axiom-Linear_Object.lean
@@ -14,7 +14,7 @@ Ray.lean -- Define (directed) segments and rays
     SEG_nd = SegND.mk -- Rewrite the standard function of making a nondegenerate segment; i.e. for (A B : P) and (nd : B ≠ A), SEG_nd A B nd gives the nondegenerate segment with source A and target B.
     Ray.mk_pt_pt -- Given two points (A B : P) and that they are not equal (nd : B ≠ A), Ray.mk_pt_pt A B nd returns the ray starting from A in the direction of B, i.e. (SEG_nd A B nd).toRay.
     RAY = Ray.mk_pt_pt -- Rewrite the making function of a ray from points (A B : P) and (nd : B ≠ A).
-  
+
   Coersion :
     Ray.toProj -- Given a (ray : Ray P), ray.toProj returns the Proj of a ray (defined from Ray.toDir)
     Ray.IsOn -- Given (a : P) (ray : Ray P), IsOn a ray is the property that point lies on a ray, namely, ∃ (t : ℝ), 0 ≤ t ∧ VEC ray.source a = t • ray.toDir.toVec.
@@ -98,7 +98,7 @@ Ray.lean -- Define (directed) segments and rays
     Seg.midpt_lies_on -- midpoint of a segment lies in its carrier
     SegND.midpt_lies_int -- if a segment is nondegenerate, the midpoint lies in its interior
     midpt_iff_same_toVec_source_and_target -- a point is the midpoint of a segment iff (SEG l.source p).toVec = (SEG p l.target).toVec
-    
+
 
     dist_target_eq_dist_source_of_midpt -- the midpoint of a segment has equal distance to the source and the target
     is_midpoint_iff_in_seg_and_dist_target_eq_dist_source -- a point is the midpoint of a segment if and only if it lies on the segment and it has same distance to the source and target
@@ -116,13 +116,13 @@ Colinarity.lean -- Define the relative positions of points on rays
 
   Definitions :
     (defn) collinear_of_nd : Prop -- Given three distinct points (A B C : P), return whether the three points are collinear, i.e. whether (VEC A B).toProj = (VEC A C).toProj
-    (defn) collinear : Prop -- Given three points (A B C : P), return whether they are collinear; if at least two of them are equal, then they are considered automatically collinear.
-  
+    (defn) Collinear : Prop -- Given three points (A B C : P), return whether they are collinear; if at least two of them are equal, then they are considered automatically collinear.
+
   Theorems :
     collinear_of_vec_eq_smul_vec -- Given three points (A B C : P), if VEC A C = t ⬝ VEC A B for some t ∈ ℝ, then A B C are collinear
     collinear_of_vec_eq_smul_vec' -- same as collinear_of_vec_eq_smul_vec, excepted stated in the form of existence of t
     collinear_iff_eq_smul_vec_of_ne -- Given three points (A B C : P) with (B ≠ A), then A B C are collinear if and only if VEC A C = r ⬝ VEC A B for some r ∈ ℝ
-    
+
 
 
 -/
@@ -130,14 +130,14 @@ Colinarity.lean -- Define the relative positions of points on rays
 /-!
 Line.lean -- Define lines
 
-  Setoid : 
+  Setoid :
     same_extn_line -- the equivalent relation of admitting the same underlying line on rays
 
-  Classes : 
+  Classes :
     class of Line -- define Line as the quotient of Ray with respect to relation same_extn_line
 
   Make :
-    LIN = Line.mk_pt_pt -- 
+    LIN = Line.mk_pt_pt --
     Line.mk_pt_proj --
     Line.mk_pt_dir --
     Line.mk_pt_vec_nd --
@@ -163,7 +163,7 @@ Line.lean -- Define lines
       snd_pt_lies_on_line_of_pt_pt --
       eq_line_of_pt_pt_of_ne --
       eq_of_pt_pt_lies_on_of_ne -- 2 ne pts lies on 2 lines implies 2 lines are the same
-    coercion : theorems related to coercions 
+    coercion : theorems related to coercions
       SegND.toLine_eq_toRay_toLine --
       line_of_pt_pt_eq_ray_toLine --
       line_of_pt_pt_eq_seg_nd_toLine --
@@ -191,5 +191,5 @@ Line.lean -- Define lines
     lies_on_of_SegND_toProj_eq_toProj --
     lies_on_iff_eq_toProj_of_lies_on --
     Line.exist_pt_beyond_pt --
-  
--/
\ No newline at end of file
+
+-/