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Topic: rational n-gon inscribed in a unit circle
Replies: 59   Last Post: Dec 19, 2013 12:56 AM

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 quasi Posts: 12,067 Registered: 7/15/05
Re: rational n-gon inscribed in a unit circle
Posted: Dec 11, 2013 5:21 PM

quasi wrote:
>quasi wrote:
>>
>>I'll start with the previous question, recast as a conjecture,
>>along with some related conjectures.
>>
>>Call an n-gon rational (or edge-rational to be more precise)
>>if all edge lengths are rational.
>>
>>Conjecture (1):
>>
>>If n > 6, there does not exist a rational n-gon which can be
>>inscribed in a unit circle.
>>
>>Conjecture (2):
>>
>>If a rational hexagon is inscribed in a unit circle, then
>>it's a regular hexagon.
>>
>>Conjecture (3):
>>
>>If a rational pentagon is inscribed in a unit circle, then
>>a = b = c = 1 and d^2 + e^2 = 4 for some permutation
>>a,b,c,d,e of the edge lengths.
>>
>>Conjecture (4):
>>
>>If a rational quadrilateral is inscribed in a unit circle,
>>then a^2 + b^2 = 4 and c^2 + d^2 = 4 for some permutation
>>a,b,c,d of the edge lengths.

>
>Conjecture (4) [revised]:
>
>If a rational quadrilateral is inscribed in a unit circle,
>then either
>
> a = b = c = 1 and d = 2
>
>or
>
> a^2 + b^2 = 4 and c^2 + d^2 = 4
>
>for some permutation a,b,c,d of the edge lengths.
>

>>Remarks:
>>
>>I'm not very confident of the truth of any of the above
>>conjectures.
>>
>>In the quest for proofs or disproofs, Conjecture (4) might
>>be a good place to start.

Conjecture (4) is false.

I've found lots of counterexamples.

Perhaps the spirit of the conjecture can be saved by weakening
the conclusion, but I'm on my way out, so I'll have to leave
it for now -- I'll take another look at it later tonight or
tomorrow.

I haven't yet looked for counterexamples to conjectures
(1),(2),(3), but the failure of conjecture (4) suggests that
those conjectures are probably also false. We'll see.

quasi

Date Subject Author
12/10/13 quasi
12/10/13 ross.finlayson@gmail.com
12/10/13 quasi
12/11/13 quasi
12/11/13 quasi
12/11/13 quasi
12/12/13 quasi
12/12/13 Helmut Richter
12/12/13 quasi
12/11/13 scattered
12/11/13 quasi
12/11/13 fom
12/11/13 fom
12/11/13 quasi
12/11/13 fom
12/11/13 Richard Tobin
12/11/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Brian Q. Hutchings
12/13/13 quasi
12/13/13 Brian Q. Hutchings
12/12/13 Thomas Nordhaus
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 Richard Tobin
12/12/13 quasi
12/12/13 quasi
12/13/13 Richard Tobin
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/13/13 quasi
12/12/13 Richard Tobin
12/13/13 quasi
12/13/13 Richard Tobin
12/13/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 quasi
12/14/13 Richard Tobin
12/15/13 quasi
12/15/13 quasi
12/15/13 Richard Tobin
12/15/13 David Bernier
12/15/13 quasi
12/18/13 Richard Tobin
12/18/13 ross.finlayson@gmail.com
12/19/13 quasi
12/14/13 Richard Tobin
12/14/13 quasi
12/14/13 Richard Tobin
12/14/13 ross.finlayson@gmail.com
12/15/13 Brian Q. Hutchings