I have the following problem that I'm unsure how to approach. It reads, let $P$ be an $n$-sided regular polygon such that every diagonal of $P$ lies inside $P$. count the number of pairs of diagonals of $P$ that cross.
It seems that there should be some pattern here similar to patterns for the number of diagonals, $\tfrac12 n(n-3)$, but after looking over a few basic examples, none has come to mind. Could someone please help me get started?
Best Answer
A start: (and nearly a finish) Choose any $4$ vertices. Then exactly one pair of the diagonals determined by these vertices meets inside the polygon.
Note that this counts the number of intersecting pairs, and not the number of intersection points. That problem is quite complicated, because of symmetries.