quasi
Posts:
11,877
Registered:
7/15/05


Re: rational ngon 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 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.
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

