The Hagge Circle
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Let ABC be a triangle and H its orthocenter. Let P be a point and AP, BP, and CP meet the circumcircle of ABC again at A', B', and C'. Let A'', B'', and C'' be the reflections of A', B', and C' about BC, AC, and AB. Let A''P, B''P, and C''P intersect AH, BH, and CH at X, Y, and Z. Then H, A'', B'', C'', X, Y, and Z are concyclic.
Contributed by: Jay Warendorff (March 2011)
Open content licensed under CC BY-NC-SA