Undergrad Find the Number of Triangles

[bob012345]

TL;DR

Find the number of triangles in this busy figure.

The game is to find the number of triangles in this complicated figure by other than brute force counting and explain your method.

 

[Hill]

I'd start with marking the vertices, e.g.,

Then, I'd make a 25x25 matrix with 1s for each pair of vertices connected by straight line, e.g., (1,2), (1,5), (1,13), ..., (2,5), (2,13), ... .
Then, I'd run a counting algorithm like this:
for i=1 to 25
.......for j=1 to 25
..............if M(i,j)=1 then
.......................for k=j+1 to 25
..................................if M(i,k)=1 then
...........................................if M(j,k)=1 then count+1
At the end, I'd divide the count by 6.

There should be a better way, because I didn't use any symmetries of this specific figure.