Morley triangle

plural Morley triangles

1. geometryAny of a number of triangles described in Morley's trisector theorem, especially the equilateral triangle formed from the points of intersection of adjacent angle trisectors of a given triangle.

   Show that the sides of the Morley triangle of T (cf. 10.3.10) lie on three of those lines (notice that each vertex of the Morley triangle is the center of a cardioid lying inside T and tangent to all its sides, one of them twice.)

   2009, Marcel Berger, Geometry I, page 276

   2 more examples

   The length of a side of the Morley triangle of a given triangle T is less than one third the length of the smallest side of T. Remark. The Morley triangle of T is the equilateral triangle formed by the intersection in pairs of the angle trisectors of T.

   2013, Dragoslav S. Mitrinovic, J. Pecaric, V. Volenec, Recent Advances in Geometric Inequalities, page 276

   There are a total of 18 Morley triangles that can be constructed. One amusing article I spotted on the web was from a math enthusiast who wrote a computer program trying to illustrate the central Morley triangle we started with, but due to a bug actually trisected the exterior angles in come cases... and was surprised to produce equilateral Morley triangles anyway!

   2016, Erik Seligman, Math Mutation Classics, page 27

   Synonyms: equilateral triangle formed by intersection points of adjacent trisectors