How many triangles can you find?
This app proves that the number of triangles that can be made inside of an equilateral triangle where a b and c are the number of internal lines coming from the corresponding point and no 3 of those lines cross at a single point is:
It does this by generating an svg showing every possible triangle.
| Formula | Explanation |
|---|---|
| 1 | The base triangle |
| abc | If 3 lines can't cross at a single point, they must cross at 3 points and make a triangle  |
| 2a + 2b + 2c | Each line from a creates a point xᵃ on line A, making triangles abxᵃ and acxᵃ |
| Ta-1 + Tb-1 + Tc-1 | Every pair of lines from a creates a triangle with a segment of line A as the third line |
| g(a,b) + g(b,c) + g(a,c) | Triangles created from intersections between two lines |
| where g(a,b) | = |
| 3ab | Each line from a splits each line from b at point y, making triangles ayxb, byxa, and ayb. This is similar to 2a above, but with a third triangle created using point xa. |
| Ta-1b + Tb-1a | Every pair of lines from a creates a triangle with each line from b, and vice versa. This is the same as the triangles created along the edges above. |
- If c=0, it simplifies to ½(a+1)(b+1)(a+b+2)
- If c=0 and a=b, it simplifies to (a+1)³
- When treated as a graph with a node at each line crossing and an edge between each pair of nodes in a line:
- The number of edges is a*T(b+c+1)+T(a+1)+b*T(a+c+1)+T(b+1)+c*T(a+b+1)+T(c+1)
- Where T is the Triangle Number, i.e. T(n)=n*(n+1)/2
- The number of nodes is ab + bc + ac + a + b + c + 3
- Every line from each side crosses every line from the other sides, and adds a node on the opposite base line. Plus 3 starting nodes from the base triangle.
- If c=0, the number of triangles is edges-nodes+1
- The number of edges is a*T(b+c+1)+T(a+1)+b*T(a+c+1)+T(b+1)+c*T(a+b+1)+T(c+1)
lein run and visit localhost:3000
The four boxes are a, b, c, and o, where o is an offset to add to each angle to prevent 3 lines from crossing a single point.
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