TITLE: 21 sided regular polygon and its diagonals
QUESTION [8 upvotes]: In a $21$ sides regular polygon, how many points inside it are
  intersection of its diagonal?

I found that a polygon with $n$ sides has $\dfrac{n(n - 3)}{2}$ diagonals, but I feel this is not so useful to the problem solution. I've been trying for $3$ hours without success.  
What's the correct solution? 
This is part of a contest that is already finished (the solutions have not been released yet).

REPLY [2 votes]: The general case is solved here: http://www-math.mit.edu/~poonen/papers/ngon.pdf Although the general case is solved in a rather complicated way, with use of 'heavy casework', the result for an $n$-gon, when $n$ is prime, or is a product of two prime numbers, as it is in your case can be obtained easily studying a bit the article.