# Polygons inscribed in circles, with integer sides and integer radius

Is there a simple characterization for an integer partition $(s_1,\dots,s_k)$, such that a polygon with these sides is inscribed in a circle with integer radius?

This is what I got so far: All pythagorean triplets $(\sqrt{a^2+b^2},a,b)$ of course have this property, and if $r=\frac{1}{2}\sqrt{a^2+b^2}$ is integer, then the 5-tuple $(a,b,r,r,r)$ has this property, since the diameter is $2r$.

Note that this question is equivalent to asking when the equation $$\cos\left(\sum_{j=1}^k \arccos\left( 1 - \frac{a_j²}{2r^2}\right) \right) = 1$$ has an integer solution $r\geq 1$.

• Note the similarity with the question about existence of perfect cuboids, i.e., cuboids with integer side lengths, integer side diameters, and integer space diameters. – Per Alexandersson Sep 4 '14 at 19:08
• An equivalent statement in terms of complex numbers, for those who appreciate such things: For which natural $n$ can I find a sequence $\{z_{1\leq i \leq n}\}$ such that $|z_{i+1}-z_{i}|\in \mathbb{Q}$ for all $i$ (taken cyclically)? – Semiclassical Sep 4 '14 at 19:25