Morley's Miracle
Nikos Dergiades' proof
The proof has been published (in Greek) in the bulletin of the department (Central Makedonia of Greece) of the Greek Mathematical Society "Diastasi": Nikolaos Dergiades, "A simple geometric proof of Morley's theorem", Diastasi 1991 is. 1-2 p. 37-38 Thessaloniki-Greece.
Lemma 1
The external angle B of an isosceles (AB = AC) ΔABC is equal to 90° + A/2.
Lemma 2
The incenter I of ΔABC lies on the bisector of an angle, e.g A, and sees the opposite side BC with angle BIC = 90° + A/2 and conversely.
Proof of Morley's Theorem
Given ΔABC, with angles A, B, C. If
On the sides of an arbitrary equilateral triangle A1B1C1 we construct outwardly the triangles A'B1C1, A1B'C1, and A1B1C' such that the lines B1C' and B'C1 are symmetric with respect to the perpendicular bisector of B1C1 and also
It is obvious that the angles of the three triangles A'B1C1, A1B'C1, A1B1C' at A', B', C' are A/3, B/3, C/3, respectively.
If A2 is the intersection of B'C1 and C'B1, then the triangle A2B1C1 is isosceles and A2A1 is the bisector of
By Lemma 1, we have that x + 60 = ∠B'C1B1 = 90° + A2/2.
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 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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