A CYCLIC POLYGON PROBLEM

Abstract

A cyclic polygon is a (convex) polygon whose vertices all lie on the same circle. They have been studied since atleast the time of Euclid and continuing research has revealed many of their properties. Although not all polygons are cyclic, a cyclic polygon with a given set of side lengths has the maximum area for a polygon with these side lengths. Interestingly, for a given circle, multiple non-congruent cyclic polygons with these side lengths may exist, and therefore have this same maximum area. Students will enjoy constructing the different cyclic polygons within a circle. We calculate the number of such non-congruent cyclic polygons, a precise result we could not find in the literature, but one which follows after we show its equivalence to a well-known bracelet problem. However, even here, our case-by-case solution to the bracelet problem that essentially uses only basic counting principles accessible to a large audience follows a much different path than the ones using results from abstract algebra employed by Polya and Burnside.

Received: February 22, 2023 
Accepted: March 29, 2023