Morley's Miracle
Bankoff's proof
This proof has appeared in Mathematics Magazine, 35 (1962) 223224.
In the diagram,
(1) 

(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 300°, or
Contact Front page Contents Geometry
Copyright © 19962018 Alexander Bogomolny