Geometric constructibility of Thalesian polygons

Abstract

A cyclic polygon is a convex n-gon inscribed in a circle. If, in addition, one of its sides is a diameter of the circle, then the polygon will be called Thalesian. Up to permutation, a Thalesian n-gon is determined by the lengths of its non-diametric sides. It is also determined by the distances of its non-diametric sides from the center of its circumscribed circle. We prove that the Thalesian n-gon in general can be constructed with straightedge and compass neither from these lengths if n ≥ 4, nor from these distances if n ≥ 5.

An analogous statement for the constructibility of cyclic n-gons from the side lengths was found by P. Schreiber in 1993; his statement was first proved by the present author and Á. Kunos in 2015. The 2015 paper could only prove the non-constructibility of cyclic n-gons from the distances for n even; here we extend this result for all n ≥ 5.