Morley triangle

Named for Anglo-American mathematician Frank Morley.

Morley triangle (plural Morley triangles)

1. (geometry) Any 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.
   • 2009, Marcel Berger, Geometry I‎, page 276:

     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.)

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

     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.

   • 2016, Erik Seligman, Math Mutation Classics‎, page 27:

     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!

• (equilateral triangle formed by intersection points of adjacent trisectors): first Morley triangle, Morley's triangle

• second Morley triangle
• third Morley triangle

• Morley centre / Morley center

• Hofstadter point