On the design of a simple proof for Morley’s Theorem

Abstract

The general opinion — with which I concur — is that Frank Morley’s Theorem about the angle trisectors of a triangle is a geometrical curiosity that is of historical interest at best. Independently of the (in) significance of the theorem proved by it, a proof may deserve our attention, for instance, by virtue of its structure, its simplicity, or its brevity. For the proof to be shown below, such a claim can be made: requiring no auxiliary lines or points, it is so simple that the theorem’s late discovery (1899) and the elapsed decade before the first proofs were published (1909) become the more striking. When I found this proof years ago, I was only too willing to ascribe that discovery to my great ingenuity and all that. The purpose of this note, however, is to show how, in the mean time, the art and science of proof design have advanced to a stage in which the design of such proofs has almost become a routine exercise, requiring the usual care in arrangement and notation, but a minimum of invention.