quasi
Posts:
12,027
Registered:
7/15/05


Re: rational ngon inscribed in a unit circle
Posted:
Dec 13, 2013 5:29 PM


Richard Tobin) wrote: >quasi wrote: >> >>Call an ngon rational if all edge lengths are rational. >> >>For n > 6, does there exist a rational ngon 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 ngon inscribed in a unit circle "primitiverationalunitcyclic" (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 ngon with n > 4, although I'm pretty sure such examples exist.
quasi

