Richard Tobin) wrote: >quasi wrote: >> >>Call an n-gon rational if all edge lengths are rational. >> >>For n > 6, does there exist a rational n-gon which can be >>inscribed in a unit circle? > >Lowest denominator heptagon: > > radius 40, 7 sides: 10 10 10 45 45 48 64
However it violates two of the restrictions imposed in later revisions:
(1) For any reordering of the edges, no two vertices are allowed to be diametrically opposite.
(2) The edges lengths are requied to be pairwise distinct.
A tentative definition ...
For this discussion, for n > 3, call a rational n-gon inscribed in a unit circle "primitive-rational-unit-cyclic" (PRUC) if
(1) No edge has length 2.
(2) The edge lengths are pairwise distinct.
(3) For any reordering of the edges, no diagonal has rational length.
An example of a PRUC quadrilateral (rescaled so that the radius and all edge lengths are integers) is the cyclic quadilateral found by Richard Tobin with sides 8,36,57,62 and radius 32.
I haven't yet seen an example of a PRUC n-gon with n > 4, although I'm pretty sure such examples exist.