A cyclic hexagon is drawn with all of its interior diagonals. All the intersections inside the hexagon are between exactly two diagonals.

Show that within the hexagon there is one embedded triangle that is formed by segments of the diagonals with all of the vertices formed by the intersections of the diagonals.

Generalize to a n-sided polygon (n>=6) and write a formula to count the number of possible embedded triangles.

Using Geometers' Sketchpad, the 1 triangle from the hexagon and 7 triangles from the heptagon seem apparent. Trying for the octagon, it gets very difficult to count the triangles as internal triangles start to share sides with their containing triangles. That makes it difficult to even try to put the first few numbers into the OEIS.

Posted by Charlie on 2013-11-16 11:39:19