quasi wrote: >quasi wrote: >>quasi wrote: >>> >>>A revised definition ... >>> >>>Call an n-gon "rational" if all edge lengths are rational. >>> >>>Call an n-gon "rational-unit-cyclic" if it's rational and >>>can be inscribed in a unit circle. >>> >>>Call an n-gon "primitive-rational-unit-cyclic" (PRUC) if it's >>>rational-unit-cyclic and, for any reordering of the edges, no >>>diagonal has rational length. >>> >>>Note: In my previously posted tentative definition, there were >>>additional requirements, namely: n > 3, pairwise distinct edge >>>lengths, and no edge length equal to 2. Those additional >>>requirements have now been dropped. >>> >>>Any rational-unit-cyclic triangle is PRUC, and the class of all >>>such triangles can be represented parametrically. >>> >>>There exist PRUC quadrilaterals. For example, the quadrilateral >>>found by Richard Tobin with sides 8,36,57,62 and radius 32 is >>>a PRUC quadrilateral scaled by a factor of 32. >>> >>>There are other PRUC quadrilaterals as well, but there's no >>>clear pattern that I can see relating the numerical values of >>>the edge lengths. It's not clear how to generate them except by >>>brute force search. >>> >>>So that's the question. >>> >>>For some n > 3, either general or specific, is there some >>>subclass of the class of PRUC n-gons which can be generated >>>by a method other than brute force search? Perhaps a parametric >>>representation or a recursion? >>> >>>In the meantime, the current search methods might help by >>>revealing a pattern for some subclass of PRUC n-gons. >> >>I haven't yet found a PRUC pentagon. >> >>I have my search program running, but so far, nothing. > >Still nothing, so based on that, I'll revive one of my previous >conjectures for the special case n = 5 ... > >Conjecture: > >If a pentagon inscribed in a unit circle has rational edge >lengths, then for some reordering of the edges within the >circle, two vertices are diametrically opposite.
The above conjecture wasn't the claim I intended -- I posted it too quickly, Not that I have a counterexample yet, just that I intended a weaker conclusion.
But I'm pretty sure I _can_ find a counterexample to the above conjecture using another method, not brute force search. It will still need some programming, and I don't have time to do the coding right now, but I'll give it a try tomorrow.
Here's the conjecture I intended to pose ...
There does not exist a PRUC pentagon.
If a pentagon inscribed in a unit circle has rational edge lengths, then for some reordering of the edges within the circle, some diagonal has rational length.
This revised conjecture may fail, but it should be a lot harder to find a counterexample.