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Morley's Theorem
© Copyright 2000, Jim Loy
If you trisect the angles of a triangle (impossible
with compasses and straightedge, as in Trisection Of An
Angle), they meet (as shown in the diagram) at three points which are the
vertices of an equilateral triangle. This elegant result was not known to the
ancient Greeks, and was discovered by Frank Morley in 1899. An interesting
thing about this simple result is that it is rather difficult to prove. One of
the interesting side results of some of the proofs is that the side of the
equilateral triangle is equal to 8R sin(A/3) sin(B/3) sin(C/3), where A,
B, and C are the angles of the larger triangle, and R is the radius of the
circumcircle (The Centers of a Triangle).
See Wrong Triangle in my Puzzle pages.
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