# Some properties of a regular 13-gon

- Author:
- Zoltán Kovács

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 (2

*n*-1)-gon (eventually a star-polygon),*n*≥2. Let its vertices be labeled by numbers 0, 1, 2, ..., 2*n*-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*P*0 will be of the same length as the side of the regular polygon, that is,*P*01 is an isosceles triangle. (This statement can be easily proven by computing the external angles of a polygon.)