quasi
Posts:
10,665
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 10, 2013 11:45 PM


I'll start with the previous question, recast as a conjecture, along with some related conjectures.
Call an ngon rational (or edgerational to be more precise) if all edge lengths are rational.
Conjecture (1):
If n > 6, there does not exist a rational ngon 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.
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

