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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 12:05 AM

quasi wrote:
>
>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.

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