Morley's trisector theorem

In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the Morley triangle. The theorem was discovered in 1899 by Anglo-American mathematician Frank Morley. It has various generalizations, in particular if all of the trisectors are intersected one obtains four other equilateral triangles.

Morley's trisector theorem is of interest because, as there is no Euclidean construction for trisecting an angle, there is no such construction of Morley's equilateral triangle.

Proofs

There are several proofs to Morley's theorem, many of which are in depth. Most of these proofs begin with an equilateral triangle and prove that a triangle may be built around it which will be similar to any selected triangle. More direct proofs are also available.

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:

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 triangle that is also central triangle is called the second Morley triangle and is given by these vertices:

A-vertex = 1 : 2 cos(C/3 − 2π/3) : 2 cos(B/3 − 2π/3)

B-vertex = 2 cos(C/3 − 2π/3) : 1 : 2 cos(A/3 − 2π/3)

C-vertex = 2 cos(B/3 − 2π/3) : 2 cos(A/3 − 2π/3) : 1

The third of Morley's 18 equilateral triangles that is also central triangle is called the third Morley triangle and is given by these vertices:

A-vertex = 1 : 2 cos(C/3 − 4π/3) : 2 cos(B/3 − 4π/3)

B-vertex = 2 cos(C/3 − 4π/3) : 1 : 2 cos(A/3 − 4π/3)

C-vertex = 2 cos(B/3 − 4π/3) : 2 cos(A/3 − 4π/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 others, is obtained by rotating the circumtangential triangle π/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)

Related triangle centers

The centroid of the first Morley triangle is given by

Morley center = X(356) = cos(A/3) + 2 cos(B/3)cos(C/3) : cos(B/3) + 2 cos(C/3)cos(A/3) : cos(C/3) + 2 cos(A/3)cos(B/3)

The first Morley triangle is perspective to triangle ABC, and the perspector is the point

1st Morley-Taylor-Marr center = X(357) = sec(A/3) : sec(B/3) : sec(C/3)

References

• Richard K. Guy, "The Lighthouse Theorem, Morley & Malfatti - A Budget of Paradoxes," American Mathematical Monthly 114 (2007) 97-141.
• C. O. Oakley and J. C. Baker, "The Morley trisector theorem," American Mathematical Monthly 85 (1978) 737-745.
• F. Glanville Taylor and W. L. Marr, "The six trisectors of each of the angles of a triangle," Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 119-131.

External links

• Morley's theorem - Introduction with animation by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Morley's Miracle — Several proofs of Morley's theorem at cut-the-knot
• Modern Mathematical Milestones: Morley's Mystery by Richard L. Francis (pdf)
• Morleys Theorem at MathWorld
• Morley's Trisection Theorem at MathPages
• Morley's Theorem by Oleksandr Pavlyk, The Wolfram Demonstrations Project.