Morley's Theorem, a Proof
Brian Stonebridge (University of Bristol, UK)
January 30, 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. We give here a direct Euclidean proof.
Using the notation in the diagram, since, in ΔABC,
(1)  α + β + γ = π/3. 
From triangles BPC, CQA, ARB,
(2) 

Observe that, in ΔBPC, X is the intersection of the bisectors of angles at B and C. It follows that PX is the bisector of the angle at P. Thus,
(3) 

This result holds for any other pair of diagonals PX, QY, RZ.
Assume that the diagonals are not concurrent, and thus form a triangle.
Select a diagonal, RZ, say, and, through the opposite vertex v, draw rz, parallel to RZ, with z, r on QA, RB. Then, Δzvq, Δxvq are congruent (ASA). Choose y, p so that Δxvr, Δyvr are congruent (ASA) making
(4)  vz = vx = vy, 
and ∠vpy = ∠OPY so that Δvpy, Δvpz are congruent (SAA). Then the angles of pzqxry, PZQXRY are the same.
Joining opposite sides of pzqxry we obtain Δabc, with angles 3α, 3β, 3γ and its trisectors which generate Δxyz.^{**} Triangles
** Informally," the mapping pzqxry ↔ abc is 11" suffices, but more formally the derivation of this result is as follows:
The intersections of opposite sides of pzqxry give Δabc. Since
Morley's Miracle
On Morley and his theorem
 Doodling and Miracles
 Morley's Pursuit of Incidence
 Lines, Circles and Beyond
 On Motivation and Understanding
 Of Looking and Seeing
Backward proofs
 J.Conway's proof
 D. J. Newman's proof
 B. Bollobás' proof
 G. Zsolt Kiss' proof
 Backward Proof by B. Stonebridge
 Morley's Equilaterals, Spiridon A. Kuruklis' proof
 J. Arioni's Proof of Morley's Theorem
Trigonometric proofs
 Bankoff's proof
 B. Bollobás' trigonometric proof
 Proof by R. J. Webster
 A Vectorbased Proof of Morley's Trisector Theorem
 L. Giugiuc's Proof of Morley's Theorem
 Dijkstra's Proof of Morley's Theorem
Synthetic proofs
 Another proof
 Nikos Dergiades' proof
 M. T. Naraniengar's proof
 An Unexpected Variant
 Proof by B. Stonebridge and B. Millar
 Proof by B. Stonebridge
 Proof by Roger Smyth
 Proof by H. D. Grossman
 Proof by H. Shutrick
 Original Taylor and Marr's Proof of Morley's Theorem
 Taylor and Marr's Proof  R. A. Johnson's Version
 Morley's Theorem: Second Proof by Roger Smyth
 Proof by A. Robson
Algebraic proofs
Invalid proofs
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