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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

(2) sin(3a) = 4sin(a)sin(p/3 + a)sin(p/3 - a)

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 300o, or QPR = 60o. The other two angles are similarly shown to be 60o.


Morley's Miracle

  1. 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

28715685Page copy protected against web site content infringement by Copyscape


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