Chapter 40: Circumscribing Polygons: Situation 34 From the MACMTL-CPTM Situations Project

In a geometry class, after a discussion about circumscribing circles about triangles, a student asked, “Can you circumscribe a circle about any polygon?”

A polygon that can be circumscribed by a circle is called a cyclic polygon.¹ Not every polygon is cyclic, but there are infinitely many different cyclic polygons. This can be understood by considering a given circle and all the possibilities of how many points can be placed on the circle, and then connected to form a polygon. However, there are certain classes of polygons that are noteworthy because they are always cyclic. The conditions under which a circle circumscribes a given polygon are dependent upon the relationships among the angles, the sides, and the perpendicular bisectors of the sides of the polygon. The following Foci describe classes of cyclic polygons in order of the number of their sides: triangles, certain quadrilaterals, and regular polygons. Focus 3 provides one way to check whether a given polygon is cyclic: A polygon is cyclic if and only if the perpendicular bisectors of all its sides are concurrent. Although the inclusion of various geometries would provide interesting discussion, the Foci in this Situation are limited to Euclidean geometry in a plane.