diff --git a/docs/src/Geometry/01 Triangles.md b/docs/src/Geometry/01 Triangles.md index 0c98345..26e087d 100644 --- a/docs/src/Geometry/01 Triangles.md +++ b/docs/src/Geometry/01 Triangles.md @@ -2,6 +2,13 @@ [Triangles](https://mathworld.wolfram.com/Triangle.html) are three-sided polygons that form the foundation of much geometric and trigonometric study. + + ## Types of Triangles ### By Side Length @@ -10,12 +17,140 @@ - **[Isosceles](https://mathworld.wolfram.com/IsoscelesTriangle.html):** Two sides equal - **[Scalene](https://mathworld.wolfram.com/ScaleneTriangle.html):** All sides different +```@raw html + + + + + + + + + + + + + + + + Equilateral + (all sides equal) + + + + + + + + + + + + + + + + + Isosceles + (two sides equal) + + + + + + + + + a + b + c + + + Scalene + (all sides different) + + +``` + ### By Angle Measure - **[Acute](https://mathworld.wolfram.com/AcuteTriangle.html):** All angles less than 90° - **[Right](https://mathworld.wolfram.com/RightTriangle.html):** One angle equals 90° - **[Obtuse](https://mathworld.wolfram.com/ObtuseTriangle.html):** One angle greater than 90° +```@raw html + + + + + + + + + + 60° + + + + 60° + + + + 60° + + + Acute + (all angles < 90°) + + + + + + + + + + 90° + + + + + 45° + + + + 45° + + + Right + (one angle = 90°) + + + + + + + + + + 120° + + + + + 40° + + + + 20° + + + Obtuse + (one angle > 90°) + + +``` + ## Triangle Properties ### Fundamental Properties of Triangles @@ -50,6 +185,69 @@ A [cevian](https://mathworld.wolfram.com/Cevian.html) is a line segment that joi - **[Altitude](https://mathworld.wolfram.com/Altitude.html):** Cevian perpendicular to opposite side - **[Angle bisector](https://mathworld.wolfram.com/AngleBisector.html):** Cevian that bisects the angle at a vertex +```@raw html + + + + + + + + + + + + M (midpoint) + + + + + + + Median + + + + + + + + + + + + + + + + H (foot) + + + Altitude + + + + + + + + + + + + + + + + + D + + + Angle Bisector + + +``` + ### Ceva's Theorem [Giovanni Ceva](https://en.wikipedia.org/wiki/Giovanni_Ceva) (1647-1734) was an Italian mathematician who discovered this fundamental theorem about concurrent cevians. His work laid important groundwork for projective geometry and triangle geometry. @@ -74,10 +272,150 @@ For more information, see: [Menelaus' Theorem - Wolfram MathWorld](https://mathw - **[Circumcenter](https://mathworld.wolfram.com/Circumcenter.html):** The point equidistant from all vertices, the center of the circumcircle. - **[Perpendicular bisectors](https://mathworld.wolfram.com/PerpendicularBisector.html):** The lines passing through the midpoint of each side of a triangle and which are perpendicular to the given side. The three perpendicular bisectors intersect at the circumcenter. - **[Incircle](https://mathworld.wolfram.com/Incircle.html):** The circle that is tangent to all three sides of a triangle. -- **[Incenter](https://mathworld.wolfram.com/Incenter.html):** The interior point for which distances to the sides of the triangle are equal. The three angle bisectors intersect at the incenter. +- **[Incenter](https://mathworld.wolfram.com/Incenter.html):** The center of the incircle. Hence it is also the interior point for which distances to the sides of the triangle are equal. The three angle bisectors intersect at the incenter. - **[Centroid](https://mathworld.wolfram.com/TriangleCentroid.html):** The point where the three triangle medians intersect, also known as the center of mass or barycenter. - **[Orthocenter](https://mathworld.wolfram.com/Orthocenter.html):** The point where the altitudes of a triangle intersect. +```@raw html + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + O + + + + + + Circumcircle/Circumcenter + + + + + + + + + + + + + + + + + + I + + + + + + Incenter + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + G + + + Centroid + + + + + + + + + + + + + + + + + + H + + + Orthocenter + + +``` + ## Area Calculations ### Standard Formula @@ -110,6 +448,42 @@ Right triangles have special properties and are fundamental to trigonometry. - **[Right Triangle](https://mathworld.wolfram.com/RightTriangle.html):** A triangle with one angle equal to 90°. - **[Cathetes and Hypotenuse](https://mathworld.wolfram.com/RightTriangle.html):** The two sides forming the right angle are called the cathetes or catheti, and the side opposite the right angle is called the hypotenuse. - **[Pythagorean Theorem](https://mathworld.wolfram.com/PythagoreanTheorem.html):** For a right triangle with legs $a$, $b$ and hypotenuse $c$: $$a^2 + b^2 = c^2$$ + +```@raw html +
+ + + + + + + + + + + + + + + + + + + a + b + c + + + a² + b² + c² + + + a² + b² = c² + +
+``` + - **[Pythagorean Triples](https://mathworld.wolfram.com/PythagoreanTriple.html):** a triple of positive integers a, b, and c such that a right triangle exists with legs $a$,$b$ and hypotenuse $c$. This is equivalent to finding positive integers $a$,$b$ and $c$ satisfying the Pythagorean Theorem: - **Examples:** - 1: $(3, 4, 5)$ @@ -143,6 +517,67 @@ Right triangles have special properties and are fundamental to trigonometry. ### Special Right Triangles +```@raw html +
+ + + + + + + + + + + + + + + + a + a + a√2 + + + 45° + 45° + + + + 45°-45°-90° Triangle + Ratio: 1 : 1 : √2 + + + + + + + + + + + + + + + + + a√3 + a + 2a + + + 60° + 30° + + + + 30°-60°-90° Triangle + Ratio: 1 : √3 : 2 + +
+``` + #### [Isosceles Right Triangle](https://mathworld.wolfram.com/IsoscelesRightTriangle.html) - Angles are 45°-45°-90°