quasi
Posts:
11,740
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 11, 2013 12:05 AM


quasi wrote: > >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.
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

