Morley's Theorem, a Proof

Brian Stonebridge 2009

In the diagram below, the near trisectors of the internal angles at the vertices $A,$ $B,$ and $C$ of a triangle meet in $X,$ $Y,$ and $Z.$ Morley's theorem states that the triangle $XYZ$ is equilateral.

Morley's theorem, Brian Stonebridge's backward proof, 1

Using the notation in the diagram, since, in $\Delta ABC, $3\alpha + 3\beta + 3\gamma = \pi ,$

$\alpha + \beta + \gamma = \pi /3.$

Start with an arbitrary equilateral triangle $XYZ.$.

Let $P,$ $Q,$ $R$ be point on the altitudes $\Delta XYZ$ (produced) such that

$\displaystyle\angle XPY=\angle XPZ=\alpha+\frac{\pi}{6},\\ \angle YQZ=\angle YQX=\beta+\frac{\pi}{6},\\ \angle ZRX=\angle ZRY=\gamma+\frac{\pi}{6}. $

Morley's theorem, Brian Stonebridge's backward proof, 2

Define $A$ to be the intersection of $QZ$ and $RY,$ $B$ the intersection of $RX$ and $PZ,$ and $C$ that of $PY$ and $QX.$ Then in quadrilateral $XRAQ,$ $\displaystyle\angle AQX=2\beta +\frac{\pi}{3},$ $\displaystyle\angle ARX=2\gamma +\frac{\pi}{3},$ and $\displaystyle\angle QXR=2\alpha +\beta +\gamma+\frac{\pi}{3}.$ It follows that $\angle ZAY=\alpha.$ Similarly, $\angle XBZ =\beta$ and $\angle YCX=\gamma.$

Draw circle with center $X$ touching $PB$ and, since $PX$ bisects $\angle BPC,$ it also touches $PC.$ Next, draw tangents $BT$ and $CU$, set $V$ as the intersection of the two lines. Then,

$\angle XBT=\angle XBZ=\beta$ and $\angle XCU =\angle XCY=\gamma.$

Now, the sum of angles $P,$ $B,$ and $C$ in quadrilateral $PBVC$ equals

$\displaystyle\angle QXR=2\alpha+\frac{\pi}{3} +2\beta +2\gamma=\pi,$

implying that $\angle TVU=0.$ In other words, $BTVUC$ is a straight line so that $T$ and $U$ coincide in $V.$ It follows that $\angle XBC=\beta$ and $\angle XCB=\gamma.$ In the same manner the angles of triangles $YCA$ and $ZAB$ are determined, letting one to conclude that triangle $ABC$ has angles $3\alpha,$ $3\beta,$ and $3\gamma.$ It may be scaled (if need be) to coincide with the original triangle.

  1. B. Stonebridge, A Simple Geometric Proof of Morley's Trisector Theorem, Applied Probability Trust, 2009

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

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