Pompeiu's Theorem Revisited

Abstract

A quick proof, by construction, of the first part of this result goes as follows: consider a rotation of π/3 radians about the point C that maps point A to point B, and M to M . Clearly, △MCM ′ is equilateral and MM ′ = MC. Since MA = M B, we conclude that △MBM ′ has sides equal to MA,MB, and MC (see Figure 1; all marked angles are equal to π/3 radians). The theorem is not classical in that not every geometry textbook mentions it. Nowadays, it is more likely that a student will encounter this theorem as a corollary of the so-called Ptolemy inequality, named after Claudius Ptolemy, who is credited with the proof of the corresponding equality in a cyclic quadrilateral, that is, a quadrilateral with its four vertices on a circle. A slight drawback, however, of this approach