Geometric constructibility of cyclic polygons and a limit theorem

Abstract

We study convex cyclic polygons, that is, inscribed $n$-gons. Starting from P. Schreiber's idea, published in 1993, we prove that these polygons are not constructible from their side lengths with straightedge and compass, provided $n$ is at least five. They are non-constructible even in the particular case where they only have two different integer side lengths, provided that $n\neq 6$. To achieve this goal, we develop two tools of separate interest. First, we prove a limit theorem stating that, under reasonable conditions, geometric constructibility is preserved under taking limits. To do so, we tailor a particular case of Puiseux's classical theorem on some generalized power series, called Puiseux series, over algebraically closed fields to an analogous theorem on these series over real square root closed fields. Second, based on Hilbert's irreducibility theorem, we give a \emph{rational parameter theorem} that, under reasonable conditions again, turns a non-constructibility result with a transcendental parameter into a non-constructibility result with a rational parameter. For $n$ even and at least six, we give an elementary proof for the non-constructibility of the cyclic $n$-gon from its side lengths and, also, from the \emph{distances} of its sides from the center of the circumscribed circle. The fact that the cyclic $n$-gon is constructible from these distances for $n=4$ but non-constructible for $n=3$ exemplifies that some conditions of the limit theorem cannot be omitted.