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. Newman's proof
3. Bankoff's proof
4. Another proof
5. Nikos Dergiades' proof
6. G. Zsolt Kiss' proof
7. M. T. Naraniengar's proof
8. Doodling and Miracles
9. Morley's Pursuit of Incidence
10. Lines, Circles and Beyond
11. On Motivation and Understanding
12. Bankoff's Conundrum
13. Morley's Redux and More, Alain Connes' proof
14. An Unexpected Variant

Copyright © 1996-2008 Alexander Bogomolny