Abstract. A \emph{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 \emph{Thalesian}. Up to permutation, a Thalesian $n$-gon is determined by the \emph{lengths} of its non-diametric sides. It is also determined by the \emph{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 $n\geq4$, nor from these distances if $n\geq5$. 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 $n\geq5$.

DOI: 10.14232/actasm-015-072-8

AMS Subject Classification (1991): 51M04, 12D05

Keyword(s): inscribed polygon, cyclic polygon, circumscribed polygon, compass and ruler, straightedge and compass, Thalesian polygon

Received September 17, 2015. (Registered under 72/2015.)