Morley's Miracle
Bankoff's proof
This proof has appeared in Mathematics Magazine, 35 (1962) 223224.
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 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
Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny
71772724