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.)
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