I am having a difficulty in order to find a generalized formula to find the number of intersection points of diagonals for a regular polygon.I will be really thankful if someone please help me in this aspect?

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    $\begingroup$ Here is a paper on the subject: www-math.mit.edu/~poonen/papers/ngon.pdf $\endgroup$ – JimmyK4542 Jul 1 '14 at 5:22
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    $\begingroup$ @Jimmy, let me encourage you to post that as an answer. There won't be a better one, unless someone chooses to type out the whole Poonen-Rubinstein paper here. $\endgroup$ – Gerry Myerson Jul 1 '14 at 5:33
  • $\begingroup$ I tried to go through that paper but i didn't understand. please do me a favour just post the generalized formula because I am in urgent need of it. $\endgroup$ – user155070 Jul 1 '14 at 15:20
  • $\begingroup$ Answer. $\endgroup$ – Lucian Aug 11 '15 at 5:03

If no three diagonals had a common intersection point, we could do something like this:

Pick any four vertices. Label them $A,B,C,D$ in clockwise order. Now draw all the lines between these four points. You get precisely one intersection point that is inside the polygon, specifically $AC$ intersects $BD$. This holds for any choice of $4$ vertices, so the number of intersection points is $\dbinom{n}{4}$.

Of course, in a regular polygon, there can be several points at which three or more diagonals intersect. So, the number of intersection points will be less than $\dbinom{n}{4}$. In this paper published by Poonen and Rubinstein, they derive the formula for the number of intersection points in a regular $n$-gon. The formula is stated in Theorem 1 on page 3 of that paper.


See if this youtube video entitled Circle Division Solution helps to answer your question.


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