trisector theorem above, called the first Morley triangle, has vertices given in trilinear coordinates relative to a triangle ABC as follows: A-vertex = 1 : 2 cos(C/3) : 2 cos(B/3) B-vertex = 2 cos(C/3) : 1 : 2 cos(A/3) C-vertex = 2 cos(B/3) : 2 cos(A/3) : 1 Another of Morley's equilateral triangles that is also a central triangle is called the second Morley triangle and is given by these vertices: A-vertex = 1 : 2 cos(C/3 − 2/3) : 2 cos(B/3 − 2/3) B-vertex = 2 cos(C/3 − 2/3) : 1 : 2 cos(A/3 − 2/3) C-vertex = 2 cos(B/3 − 2/3) : 2 cos(A/3 − 2/3) : 1 The third of Morley's 18 equilateral triangles that is also a central triangle is called the third Morley triangle and is given by these vertices: A-vertex = 1 : 2 cos(C/3 − 4/3) : 2 cos(B/3 − 4/3) B-vertex = 2 cos(C/3 − 4/3) : 1 : 2 cos(A/3 − 4/3) C-vertex = 2 cos(B/3 − 4/3) : 2 cos(A/3 − 4/3) : 1 The first, second, and third Morley triangles are pairwise homothetic. Another homothetic triangle is formed by the three points X on the circumcircle of triangle ABC at which the line XX −1 is tangent to the circumcircle, where X −1 denotes the isogonal conjugate of X. This equilateral triangle, called the circumtangential triangle, has these vertices: A-vertex = csc(C/3 − B/3) : csc(B/3 + 2C/3) : −csc(C/3 + 2B/3) B-vertex = −csc(A/3 + 2C/3) : csc(A/3 − C/3) : csc(C/3 + 2A/3) C-vertex = csc(A/3 + 2B/3) : −csc(B/3 + 2A/3) : csc(B/3 − A/3) A fifth equilateral triangle, also homothetic to the others, is obtained by rotating the circumtangential triangle /6 about its center. Called the circumnormal triangle, its vertices are as follows: A-vertex = sec(C/3 − B/3) : −sec(B/3 + 2C/3) : −sec(C/3 + 2B/3) B-vertex = −sec(A/3 + 2C/3) : sec(A/3 − C/3) : −sec(C/3 + 2A/3) C-vertex = −sec(A/3 + 2B/3) : −sec(B/3 + 2A/3) : sec(B/3 − A/3) An operation called "extraversion" can be used to obtain one of the 18 Morley triangles from another. Each triangle can be extraverted in three different ways; the 18 Morley triangles and 27 extravert pairs of triangles form the 18 vertices and 27 edges of the Pappus graph. Related triangle centers The centroid of the first Morley triangle is given in trilinear coordinates by