Inequalities for Inscribed and Circumscribed Polygons

Abstract

In this note, we elaborate on Theorem III. Although it is not stated explicitly in the theorem, it is tacitly assumed from the context that the two polygons are convex and that the center of the circle lies in the interior of both polygons. Uspensky's proof is a nice illustration of induction in geometry, which is not too prevalent (for other examples of geometric induction, see [1]). The inductive step appears in the proof of Theorem I where he first considers the case when angles a and /8 are commensurable so that a =pO and /3 = q3 where p and q are relatively prime integers. It is then supposed that the above inequalities are valid in every case when the sum p + q is less than a given integer N.> 2, from which it is shown to hold forp + q = N. Then the incommensurable case is reduced to the former case by a limiting process. The proof here of Theorem III will be based on the convexity of sinx, which also leads to further related results. Let the central angles subtended by the sides of 6P and 6' from the center of the circle be denoted by 01,02, Om and 9', 02x' ... ,On,f respectively. Uspensky assumes that 0 <Os, O' < -r, and