Some properties of a regular 13-gon
A regular 13-gon is given. Diagonals AC, BF and BH are drawn. Intersection points N and O are defined as shown in the figure.
Prove that
- a=b,
- a+c=d.
A generalization
Given a regular (2n-1)-gon (eventually a star-polygon), n≥2. Let its vertices be labeled by numbers 0, 1, 2, ..., 2n-1. Draw diagonals connecting vertices 0 and 2, and 1 and (n+1), respectively. Draw the intersection point P of these diagonals. Now the segment P0 will be of the same length as the side of the regular polygon, that is, P01 is an isosceles triangle.
(This statement can be easily proven by computing the external angles of a polygon.)