Folklore

The vertices and midpoints of the sides of a regular decagon are marked (that is, a total of 20 points are marked).

How many triangles exist with vertices at the marked points?
A triangle is uniquely determined by its three vertices. From twenty points, three can be chosen in $C_{20}^{3}=$ 1140 ways. Considering that three points lying on one side of a decagon do not form a triangle, we get: $1140-10=1130$ triangles.

## Answer

1130 triangles.