Morley's Theorem
M. T. Naraniengar's proof
This proof is one of the earliest. It has been appreciatively popularized by Coxeter and Greitzer. Morley's theorem was presented as a problem in Mathematical Questions and Their Solutions sections in the Educational Times (New Series), v 15 (1909), p. 47. One of the solutions sent in was by M. T. Naraniengar and is discussed below.
Naraniengar's proof depends on the following
Lemma
If four points Y', Z, Y, Z' satisfy the conditions
 Y'Z = ZY = YZ' and
 ∠YZY' = ∠Z'YZ = 180°  2a > 60°
then they lie on a circle. Moreover, if a point A, on the side of the line Y'Z' away from point Y, is so situated that ∠Y'AZ' = 3a, then this fifth point A also lies on the same circle.
Following Coxeter and Greitzer, let the bisectors of angles YZY' and Z'YZ meet at O. Then triangles OY'Z, OZY, OYZ' are congruent and isosceles having base angles
Now for the proof of Morley's theorem. Morley's trisectors of angles
∠ZXU = ∠YXU = 30°. 
Then triangles ZXU and YXU are equal by ASA. Therefore,
In addition, ΔUZY is isosceles with base angles at Y and Z equal to
∠YZU = 60°  a and ∠XZU = 120°  a. 
Next, mark Y' on AB and Z' on AC so that
BY' = BX and CZ' = CX. 
This makes triangles BZX and BZY' congruent by SAS; and the same holds for triangles CYX and CYZ'. Thus we make a step towards the premises of Lemma:
Y'Z = ZX = ZY = YX = YZ'. 
We have to check the second condition:

Similarly, ∠Z'YZ = 180°  2a; and of course
References
 H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967
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
69098410