0
$\begingroup$

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?

$\endgroup$
  • 4
    $\begingroup$ Here is a paper on the subject: www-math.mit.edu/~poonen/papers/ngon.pdf $\endgroup$ – JimmyK4542 Jul 1 '14 at 5:22
  • 1
    $\begingroup$ @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
2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.