Morley's Miracle
Bankoff's proof

This proof has appeared in Mathematics Magazine, 35 (1962) 223-224.

In the diagram,

(1)	sin(AQC) = sin(p - (A+C)/3)   = sin((p - B)/3)   = sin((2p + B)/3)

Also

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^o, or QPR = 60^o. The other two angles are similarly shown to be 60^o.

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