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Morley's miracle




A remarkable theorem, discovered in 1899, by Frank Morley, then professor of mathematics at Haverford College. Take any triangle. Mark the three points that are the intersections of adjacent angle trisectors. Then, no matter what triangle you start with, these three points will form an equilateral triangle. That such a simple and elegant result was not known to the ancient Greeks may be because it is quite hard to prove. One of the interesting auxiliary 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. A surprise awaits anyone who takes the intersections of the exterior, as well as the interior, angle trisectors. In addition to the interior equilateral triangle, four exterior equilateral triangles appear, three of which have sides that are extensions of a central triangle.


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   • GEOMETRY