A Construction of Regular Polygons of pq Sides Leading to a Geometric Proof of rp - sq=l

Abstract

The construction of 15, 51, and 85 sided polygons is an interesting exercise when studying ruler and compass constructions. The method is based on the famous Greatest Common Divisor theorem that if p and q are relatively prime, there exist counting numbers r, s such that rp-sq= 1. The proof of this theorem, while elementary, involves descent, and working back through a messy chain of equations. The spirit of the algebraic steps is quite different from that of geometric construction. Hence it is interesting in a geometry class to use a proof less dependent on an algebraic lemma. Indeed, a proof of the lemma falls out as a corollary.