Morley's Miracle
Bankoff's proof

This proof has appeared in Mathematics Magazine, 35 (1962) 223-224.

In the diagram,

(1)
sin(∠AQC) = sin(p - (A+C)/3)
  = sin((p - B)/3)
  = sin((2p + B)/3)

Also

(2) sin(3a) = 4sin(a)sin(p/3 + a)sin(p/3 - a)

From the Sine Law,

AQ·sin((p - B)/3) = 2R·sin(B)·sin(C/3),

where R is the circumradius. Therefore, by (2)

AQ = 8R·sin(B/3)·sin(C/3)·sin((p + B)/3).

Similarly, AR = 8R·sin(C/3)·sin(B/3)·sin((p + C)/3). Therefore,

AR/AQ = sin((p + C)/3)/sin((p + B)/3).

But ∠ARQ + ∠AQR = p - A/3 = (p + B)/3 + (p + C)/3. From here,

∠ARQ = (p + C)/3 and ∠AQR = (p + B)/3,

and similarly for triangles BPR and CPQ. It thus follows that the sum of angles around P, excluding ∠QPR is 300°, or ∠QPR = 60°. The other two angles are similarly shown to be 60°.


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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