## Diagonals in Polygons

Shape and space, angle and circle properties, ...
daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 10:15 pm
Location: Bremen, Germany

### Diagonals in Polygons

Consider a convex n-gon P (all interior angles strictly less than 180&deg;).
A diagonal in P is a straight line segment connecting two of P's vertices which aren't neighbours.

What is the largest number of diagonals you can draw in P so that each of the drawn diagonals intersects at most two others (in the interior of P)?
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.

stijn263
Posts: 1505
Joined: Sat Sep 15, 2007 10:57 pm
Location: Netherlands

### Re: Diagonals in Polygons

n ans
3 0
4 2
5 5

I can always draw n diagonals in any n gon with n > 4, but is that optimal ? edit: I hope so, because I think I can prove it

daniel.is.fischer
Posts: 2400
Joined: Sun Sep 02, 2007 10:15 pm
Location: Bremen, Germany

### Re: Diagonals in Polygons

No, it's not optimal. For example, in an octogon, you can draw 11 diagonals such that each intersects at most two others.
Like that:
Octo.gif (2.18 KiB) Viewed 9611 times
Il faut respecter la montagne -- c'est pourquoi les gypa&egrave;tes sont l&agrave;.