Problem

Source: Balkan MO 2010, Problem 2

Tags: geometry, geometric transformation, reflection, circumcircle, cyclic quadrilateral, geometry proposed



Let ABC be an acute triangle with orthocentre H, and let M be the midpoint of AC. The point C1 on AB is such that CC1 is an altitude of the triangle ABC. Let H1 be the reflection of H in AB. The orthogonal projections of C1 onto the lines AH1, AC and BC are P, Q and R, respectively. Let M1 be the point such that the circumcentre of triangle PQR is the midpoint of the segment MM1. Prove that M1 lies on the segment BH1.