What is the easiest way to find the number of triangles in this picture?
Solution 1:
A triangle is formed by three lines intersecting at three different points. There are $8$ line segments in the picture, so at most $\binom83=56$ triangles. Since all line segments meet in the picture, the only way three of them can fail to form a triangle is if they all meet at one point. There are $\binom53=10$ concurrent triples at the bottom left corner and $\binom43=4$ concurrent triples at the bottom right corner, so the actual number of triangles in the picture is $$\binom83-\binom53-\binom43=56-10-4=\boxed{42}.$$