Morley's Theorem
Roger Smyth
Morley's theorem asserts that in the diagram below the triangle XYZ is equilateral, whatever ΔABC.
In the next diagram, the unit of measure is the perpendicular DX, and the lengths of BX and CX are s and s'. E and F are points on BC where
What can be said of the diagram?
- α + β + γ = 60° (so ∠EXF = α)
- ∠CYA = ∠CLP' = 120° + β
- ∠TSB = ∠SPX = 60°
- AY = (AC/s')LP' (for, ΔCP'L is similar to ΔCAY)
- LP' = PS/ST (for ∠P'LY = ∠SPT = 60° - β)
- XR = RV = 2/s (for triangles BDX, PRX, PVR are similar)
- SU = s/2 (for ∠BSU = 60°)
Now, by (vi) and (vii),
2 ST = 2 SU + 2 UT = s + 2(s - 2/s) = 3s - 4/s.
Also, since triangles BQV and BDX are similar,
PQ = PV + VQ = 3 - 4/s²,
leading to 2 ST = s PQ.
Next, by (iv) and (v),
AY = (AC / s')(PS / ST) = 2 AC·PS / (ss' PQ).
But XE = 2s/PS as ΔPBS is isosceles and ∠SBP = 2(60° + β) = 2∠DEX. It follows that
XE·AY = 4 AC / (s' PQ)
and, by symmetry,
(XE·AY) / (XF·AZ) = [4 AC / (s' PQ)] [(s P'Q') / (4 AB)] = (s AC·P'Q') / (s' AB·PQ) = 1
because PQ(AB/s) = P'Q'(AC/s') is the altitude of ΔABC. However, this means
References
- M. R. F. Smyth, MacCool's proof of Morley's Miracle, Irish Math. Soc. Bulletin 63 (2009), 63-66

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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