I. There are three ways of trisecting an angle bounded by OA and OB ( = 3αor 3α +2π or 3α +4π), namely:

(i) OX 1 and OY 1,

(ii) OX 2 and OY 2,

(iii) OX 3 and OY 3,

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.