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Orthocenter, incenter and three other points lie on a circle

Source: Iranian National Olympiad (3rd Round) 2003

January 24, 2009
geometryincentercircumcircletrapezoidgeometric transformationreflectiongeometry proposed

Problem Statement

Suppose that M M is an arbitrary point on side BC BC of triangle ABC ABC. B1,C1 B_1,C_1 are points on AB,AC AB,AC such that MB=MB1 MB = MB_1 and MC=MC1 MC = MC_1. Suppose that H,I H,I are orthocenter of triangle ABC ABC and incenter of triangle MB1C1 MB_1C_1. Prove that A,B1,H,I,C1 A,B_1,H,I,C_1 lie on a circle.
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