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Geometry ProblemLevel: Mathematics Education, High School, Honors Geometry, College.Click the figure below to see the complete problem 929.
http://img43.imageshack.us/img43/9618/h2bb.pngLet BH,AH and CH cut the circle at B’, A’ and C’ Note that A’, B’ and C’ is the reflection of H over BC, AC and AB ..( property of orthocenter)So F, B’, B1 are collinear , similarly for E, A’, A1 and D, C’, C1 This problem become problem 733 . see below for solutionhttp://gogeometry.blogspot.com/2012/02/problem-733-triangle-orthocenter.html
Let A', B', and C' be reflections of H over BC, AC, and AB respectively. By property of orthocenter, these points are on circumcircle. Let FB' and DC' meet at P. <FPD=180-2(<DFA+<FDA)=180-2<BAC=arcB'C'/2, making P on circumcircle.Let DC' and EA' meet at P'.<DP'E=180-(180-2<BDE)-(180-2<BED)=2(<BDE+<BED)-180=2(180-<ABC)-180=180-2<ABC=arcA'C'/2, making P' on circumcircle.P and P' are both on DC' and circumcircle, so they must coincide.