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 ∠CEX = 60° + β and ∠XFB = 60° + γ. P and P' are where BP = s and CP' = s' and S is constructed so that BS = s and SBX = 120°. This makes the two marked angles (TSB and SPX) 60°. The rest of the construction is hopefully as self-explanatory.

What can be said of the diagram?

Now, by (vi) and (vii),

Also, since triangles BQV and BDX are similar, VQ = 1 - 4/s² so

leading to 2 ST = s PQ.

Next, by (iv) and (v),

But XE = 2s/PS as ΔPBS is isosceles and ∠SBP = 2(60° + β) = 2∠DEX. It follows that

and, by symmetry,

because PQ(AB/s) = P'Q'(AC/s') is the altitude of ΔABC. However, this means AZ / AY = XE / XF which together with ∠ZAY = α = ∠EXF implies that triangles AZY and XEF are similar. Hence ∠YZA = ∠FEX = 60° + β and ∠AYZ = ∠XFE = 60° + γ. Analogous arguments for ΔBXZ and ΔCYX give ∠ZXB = 60° + γ, ∠CXY = 60° + β and ∠BZX = ∠CYX = 60° + α. All the angles in the diagram may now be deduced in terms of α, β, γ and it transpires that every angle of ΔXYZ is 60°.

References

1. M. R. F. Smyth, MacCool's proof of Morley's Miracle, Irish Math. Soc. Bulletin 63 (2009), 63-66

Morley's Miracle

1. J.Conway's proof
   • Remarks on J. Conway's proof
2. D. J. Newman's proof
3. Bankoff's proof
4. B. Bollobás' proof
5. Another proof
6. Nikos Dergiades' proof
7. G. Zsolt Kiss' proof
8. M. T. Naraniengar's proof
9. Doodling and Miracles
10. Morley's Pursuit of Incidence
11. Lines, Circles and Beyond
12. On Motivation and Understanding
13. Bankoff's conundrum
14. Of Looking and Seeing
15. Morley's Redux and More, Alain Connes' proof
16. An Unexpected Variant
17. Proof by B. Stonebridge and B. Millar
18. Proof by B. Stonebridge
19. Proof by Nolan L Aljaddou
20. Proof by Roger Smyth

Copyright © 1996-2009 Alexander Bogomolny