A hexagon has its three long diagonals drawn, and its alternate corners are joined to form a triangle. Count all the triangles.

Count by size, in units of the smallest piece: 12 of size 1, 12 of size 2, 6 of size 3, 6 of size 4, and the big triangle on alternate corners (size 6), 1. Total $12+12+6+6+1=37$.