Editing Morley's trisector theorem (section)

==Morley's triangles==
Morley's theorem entails 18 equilateral triangles.  The triangle described in the trisector theorem above, called the '''first Morley triangle''', has vertices given in [[trilinear coordinates]] relative to a triangle ''ABC'' as follows:

<math display=block>\begin{array}{ccccccc}
  A \text{-vertex} &=& 1 &:& 2 \cos\tfrac13 C &:& 2 \cos\tfrac13 B \\[5mu]
  B \text{-vertex} &=& 2 \cos\tfrac13 C &:& 1 &:& 2 \cos\tfrac13 A \\[5mu]
  C \text{-vertex} &=& 2 \cos\tfrac13 B &:& 2 \cos\tfrac13 A &:& 1
\end{array}</math>

Another of Morley's equilateral triangles that is also a central triangle is called the '''second Morley triangle''' and is given by these vertices:

<math display=block>\begin{array}{ccccccc}
  A \text{-vertex} &=& 1 &:& 2 \cos\tfrac13(C - 2\pi) &:& 2 \cos\tfrac13(B - 2\pi) \\[5mu]
  B \text{-vertex} &=& 2 \cos\tfrac13(C - 2\pi) &:& 1 &:& 2 \cos\tfrac13(A - 2\pi) \\[5mu]
  C \text{-vertex} &=& 2 \cos\tfrac13(B - 2\pi) &:& 2 \cos\tfrac13(A - 2\pi) &:& 1
\end{array}</math>

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:

<math display=block>\begin{array}{ccccccc}
  A \text{-vertex} &=& 1 &:& 2 \cos\tfrac13(C + 2\pi) &:& 2 \cos\tfrac13(B + 2\pi) \\[5mu]
  B \text{-vertex} &=& 2 \cos\tfrac13(C + 2\pi) &:& 1 &:& 2 \cos\tfrac13(A + 2\pi) \\[5mu]
  C \text{-vertex} &=& 2 \cos\tfrac13(B + 2\pi) &:& 2 \cos\tfrac13(A + 2\pi) &:& 1
\end{array}</math>

The first, second, and third Morley triangles are pairwise [[Homothetic transformation|homothetic]].  Another homothetic triangle is formed by the three points ''X'' on the circumcircle of triangle ''ABC'' at which the line ''XX''<sup>&nbsp;&minus;1</sup> is tangent to the circumcircle, where ''X''<sup>&nbsp;&minus;1</sup> denotes the [[isogonal conjugate]] of ''X''.  This equilateral triangle, called the '''circumtangential triangle''', has these vertices:

<math display=block>\begin{array}{lllllll}
  A \text{-vertex} &=& \phantom{-}\csc\tfrac13(C - B) &:& \phantom{-}\csc\tfrac13(2C + B) &:& -\csc\tfrac13(C + 2B) \\[5mu]
  B \text{-vertex} &=& -\csc\tfrac13(A + 2C) &:& \phantom{-}\csc\tfrac13(A - C) &:& \phantom{-}\csc\tfrac13(2A + C) \\[5mu]
  C \text{-vertex} &=& \phantom{-}\csc\tfrac13(2B + A) &:& -\csc\tfrac13(B + 2A) &:& \phantom{-}\csc\tfrac13(B - A)
\end{array}</math>

A fifth equilateral triangle, also homothetic to the others, is obtained by rotating the circumtangential triangle {{pi}}/6 about its center.  Called the '''circumnormal triangle''', its vertices are as follows:

<math display=block>\begin{array}{lllllll}
  A \text{-vertex} &=& \phantom{-}\sec\tfrac13(C - B) &:& -\sec\tfrac13(2C + B) &:& -\sec\tfrac13(C + 2B) \\[5mu]
  B \text{-vertex} &=& -\sec\tfrac13(A + 2C) &:& \phantom{-}\sec\tfrac13(A - C) &:& -\sec\tfrac13(2A + C) \\[5mu]
  C \text{-vertex} &=& -\sec\tfrac13(2B + A) &:& -\sec\tfrac13(B + 2A) &:& \phantom{-}\sec\tfrac13(B - A)
\end{array}</math>

An operation called "[[triangle extraversion|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]].<ref>{{harvtxt|Guy|2007}}.</ref>