Diagonals in Polygons

Shape and space, angle and circle properties, ...
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daniel.is.fischer
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Diagonals in Polygons

Post by daniel.is.fischer » Fri Sep 04, 2009 9:12 pm

Consider a convex n-gon P (all interior angles strictly less than 180°).
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)?
Give a proof for your answer.
Il faut respecter la montagne -- c'est pourquoi les gypaètes sont là.

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stijn263
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Re: Diagonals in Polygons

Post by stijn263 » Tue Sep 08, 2009 9:15 am

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 ;-)

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daniel.is.fischer
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Re: Diagonals in Polygons

Post by daniel.is.fischer » Tue Sep 08, 2009 9:46 am

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
Octo.gif (2.18 KiB) Viewed 3339 times
Il faut respecter la montagne -- c'est pourquoi les gypaètes sont là.

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