Morley’s Triangle

Abstract

I. There are three ways of trisecting an angle bounded by OA and OB ( = 3αor 3α +2π or 3α +4π), namely:(i) OX1 and OY1,(ii) OX2 and OY2,(iii) OX3 and OY3,as shown in Fig. 1.It is well known that if the Euclidean angles of a triangle ABC are trisected as in Fig. 2, then the triangle xyz (commonly quoted as Morley’s Triangle) is equilateral. But angles CAB, ABC, BCA may be trisected each in three ways, so that there are really 27 “Morley” triangles to be considered. The purpose of the present paper is to show that 18 of these 27 triangles are equilateral.