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Grimbal
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 Diagonals of a polygon.   « on: May 13th, 2007, 3:11pm » Quote Modify

I took part in a logical and mathematical games contest.  One of the problem was (story removed):
There is a convex hexagon such that
- all sides have a different length,
- all 3 diagonals are concurrent
- the vertices lie on the vertices of a regular N-sided polygon.
What is the minimum possible N?

As a starter, the given answer was an odd number, but in my opinion it can not be because by experience (i.e. computer), no 3 digaonals of a regular N-sided polygon are concurrent for N odd.  Does anybody know a good argument for that?
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Barukh
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 Re: Diagonals of a polygon.   « Reply #1 on: May 13th, 2007, 11:05pm » Quote Modify

on May 13th, 2007, 3:11pm, Grimbal wrote:
 - the vertices lie on the vertices of a regular N-sided polygon.

Shouldn't this be "on the sides of a regular N-gon?
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towr
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 Re: Diagonals of a polygon.   « Reply #2 on: May 14th, 2007, 12:46am » Quote Modify

on May 13th, 2007, 11:05pm, Barukh wrote:
 Shouldn't this be "on the sides of a regular N-gon?
My guess would be no. You want the diagonals of the hexagon to coincide with those of the N-gon, so it has to be vertices in both cases.
 « Last Edit: May 14th, 2007, 12:47am by towr » IP Logged

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Grimbal
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 Re: Diagonals of a polygon.   « Reply #3 on: May 14th, 2007, 2:10am » Quote Modify

As towr said.  The vertices of the hexagon are a subset of those of a regular N-gon.
 « Last Edit: May 14th, 2007, 2:43am by Grimbal » IP Logged
Barukh
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 Re: Diagonals of a polygon.   « Reply #4 on: May 14th, 2007, 8:32am » Quote Modify

I see.

Grimbal, your assumption is true. I don't know if there is a simple argument to prove it, though.

Maybe, the following is of some help.
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Grimbal
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 Re: Diagonals of a polygon.   « Reply #5 on: May 14th, 2007, 9:56am » Quote Modify

Thanks.

It shows that identifying concurrent diagonals is far from trivial and cannot be solved in the hour I had, not without prior knowledge of the problem.

But in the introduction it refers to an earlier paper:
Herman Heineken, "Regelmässige Vielecke und ihre Diagonalen", 1962.
He proves that an odd-gon has no 3 concurrent diagonals, using complex polynomials.
 « Last Edit: May 14th, 2007, 9:57am by Grimbal » IP Logged
balakrishnan
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 Re: Diagonals of a polygon.   « Reply #6 on: Aug 5th, 2007, 5:37pm » Quote Modify

I get 8 as the smallest N.
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Obob
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 Re: Diagonals of a polygon.   « Reply #7 on: Aug 5th, 2007, 10:55pm » Quote Modify

You cannot choose six vertices of an octagon in such a way that the consecutive distances between adjacent vertices are all different.
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balakrishnan
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 Re: Diagonals of a polygon.   « Reply #8 on: Aug 6th, 2007, 5:05am » Quote Modify

Sorry I overlooked the problem.
I get N=24.
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Grimbal
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 Re: Diagonals of a polygon.   « Reply #9 on: Aug 8th, 2007, 5:39am » Quote Modify

Yep.  That's what I got later with a computer program.
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