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Eight-point cicle

Source: Balkan MO 2010, Problem 2

May 4, 2010
geometrygeometric transformationreflectioncircumcirclecyclic quadrilateralgeometry proposed

Problem Statement

Let ABCABC be an acute triangle with orthocentre HH, and let MM be the midpoint of ACAC. The point C1C_1 on ABAB is such that CC1CC_1 is an altitude of the triangle ABCABC. Let H1H_1 be the reflection of HH in ABAB. The orthogonal projections of C1C_1 onto the lines AH1AH_1, ACAC and BCBC are PP, QQ and RR, respectively. Let M1M_1 be the point such that the circumcentre of triangle PQRPQR is the midpoint of the segment MM1MM_1. Prove that M1M_1 lies on the segment BH1BH_1.
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