Ptolemy's Theorem and Regular Polygons

It might be both profitable and interesting to attempt to evaluate the relative importance of some of the outstanding theorems of elementary geometry by means of a poll of mathematicians. If this were done, there is little doubt that the theorem of Pythagoras would in the opinion of most people, head the list. It is not so clear just how other important theorems would be ranked. But without doubt, Ptolemy's theorem concerning an inscribed quadrilateral is deserving of a high place. This judgemnt is warranted, both from the point of view of the use which Ptolemy made of it, and from a consideration of its power as a tool for geometric study.