Morley's Miracle,
Bella Bollobás' Trigonometric Proof

The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.

Morley's theorem - statement

For simplicity, let us write $x^{+}$ for $x+\pi/3.$ We claim that the angles of the six triangles are as in the diagram below:

Morley's theorem - solution

Note that the assignment of angles is consistent with the requirement that the sum of angles in a triangle is $\pi.$ Simple calculations show that this is indeed so and that the angles in $\Delta PQR$ are each $60^{\circ}.$ We do know that the magnitudes of the angles $AQC,$ $BPC,$ and $ARB$ are as shown. To establish Morley's theorem we need to verify that other angles (those with a single $^{+})$ are distributed as in the diagram.

We assume that the circumradius of $\Delta ABC$ equals $1/2.$ This makes $BC=\sin 3\alpha,$ $AC=\sin 3\beta,$ and $AB=\sin 3\gamma.$ With the Law of Sines, we have, in $\Delta ABR,$ $\displaystyle AR=\frac{\sin\beta}{\sin\gamma^{++}}\sin 3\gamma$ and, in $\Delta ACQ,$ $\displaystyle AQ=\frac{\sin\gamma}{\sin\beta^{++}}\sin 3\beta.$

Hence, in order that in $\Delta ARQ,$ the angles be as claimed, implying, in particular, $\displaystyle\frac{AR}{AQ}=\frac{\sin\gamma^{+}}{\sin\beta^{+}},$ we need

$\sin 3\beta\sin\gamma\sin\gamma^{+}\sin\gamma^{++}=\sin 3\gamma\sin\beta\sin\beta^{+}\sin\beta^{++}.$

But this is indeed so because

$\sin 3x=4\sin x \sin x^{+} \sin x^{++}=3\sin x-4\sin^{3}x.$


  1. B. Bollobás, The Art of Mathematics, Cambridge University Press, 2006, p. 126-127

Morley's Miracle

On Morley and his theorem

  1. Doodling and Miracles
  2. Morley's Pursuit of Incidence
  3. Lines, Circles and Beyond
  4. On Motivation and Understanding
  5. Of Looking and Seeing

Backward proofs

  1. J.Conway's proof
  2. D. J. Newman's proof
  3. B. Bollobás' proof
  4. G. Zsolt Kiss' proof
  5. Backward Proof by B. Stonebridge
  6. Morley's Equilaterals, Spiridon A. Kuruklis' proof
  7. J. Arioni's Proof of Morley's Theorem

Trigonometric proofs

  1. Bankoff's proof
  2. B. Bollobás' trigonometric proof
  3. Proof by R. J. Webster
  4. A Vector-based Proof of Morley's Trisector Theorem
  5. L. Giugiuc's Proof of Morley's Theorem
  6. Dijkstra's Proof of Morley's Theorem

Synthetic proofs

  1. Another proof
  2. Nikos Dergiades' proof
  3. M. T. Naraniengar's proof
  4. An Unexpected Variant
  5. Proof by B. Stonebridge and B. Millar
  6. Proof by B. Stonebridge
  7. Proof by Roger Smyth
  8. Proof by H. D. Grossman
  9. Proof by H. Shutrick
  10. Original Taylor and Marr's Proof of Morley's Theorem
  11. Taylor and Marr's Proof - R. A. Johnson's Version
  12. Morley's Theorem: Second Proof by Roger Smyth
  13. Proof by A. Robson

Algebraic proofs

  1. Morley's Redux and More, Alain Connes' proof

Invalid proofs

  1. Bankoff's conundrum
  2. Proof by Nolan L Aljaddou
  3. Morley's Theorem: A Proof That Needs Fixing

Related material

  • Equilateral and 3-4-5 Triangles
  • Rusty Compass Construction of Equilateral Triangle
  • Equilateral Triangle on Parallel Lines
  • Equilateral Triangle on Parallel Lines II
  • When a Triangle is Equilateral?
  • Viviani's Theorem
  • Viviani's Theorem (PWW)
  • Tony Foster's Proof of Viviani's Theorem
  • Viviani in Isosceles Triangle
  • Viviani by Vectors
  • Slanted Viviani
  • Slanted Viviani, PWW
  • Triangle Classification
  • Napoleon's Theorem
  • Sum of Squares in Equilateral Triangle
  • A Property of Equiangular Polygons
  • Fixed Point in Isosceles and Equilateral Triangles

  • |Contact| |Front page| |Contents| |Geometry| |Store|

    Copyright © 1996-2017 Alexander Bogomolny


    Search by google: