Morley's Miracle
Bankoff's proof
This proof has appeared in Mathematics Magazine, 35 (1962) 223-224.
In the diagram,
(1) |
|
(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

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 Vector-based 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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