Home > Combinatorics, Geometry > Intersections of diagonals in regular polygons

Intersections of diagonals in regular polygons

I found the following question on AoPS:

Suppose we have a regular polygon with p sides. Is it true that no three of its diagonals intersect?

After a quick search, I found the following article which solves the problem for a more general case. The article provides a formula for calculating the number of intersection points of a regular polygon’s diagonals.

Poonen, B. and Rubinstein, M. “Number of Intersection Points Made by the Diagonals of a Regular Polygon”

The formula is quite complicated, but for a prime number it reduces to I(n)=\binom{n}{4}, which is exactly the number of quadrilaterals formed by the vertices of the polygon. If three or more diagonals would meet then tne number of intersection points would be smaller than the number or quadrilaterals, which is a contradiction. Therefore every three diagonals of the polygon do not have a point in common.

I think the formula for a prime number could be derived without the general one. Using the properties presented in the first paragraphs of the article, this thing can be done.

Source of the article.

Categories: Combinatorics, Geometry
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