TITLE: Can three diagonals in a $2k+1$-gon intersect?
QUESTION [5 upvotes]: Is it possibly to find three diagonals in a regular $2k+1$-gon that intersect? More particularly can  they intersect inside the polygon? I am looking for an elementary solution. Can one be found without using trigonometry? Any solution is welcome.

REPLY [2 votes]: Great question. No, it's not possible. 
At this page http://arxiv.org/pdf/math/9508209.pdf you can find  a complete answer (for any regular $n$-gon).  It used to be a famous unsolved question of Steinhaus, for $2k+1$ prime ...