Creating Triangular Images from Parallel Lines

Part II keys off of fact that the vertices of regular polygons can be used to create parallel lines. The easiest way to see this is to look at a clock and note that segments connecting 1 and 11, 2 and 10, 3 and 9, 4 and 8, and 5 and 7 are all parallel (and horizontal). The same result can be seen in other, non-horizontal, directions (for example the segments connecting 12 and 1 is parallel to 11 and 2, 10 and 3, etc.). When one chooses three non-parallel lines using vertices of the polygon and then draws all lines parallel to those lines, the image that results includes many similar triangles. The question examined here involves how to count these triangles.