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O,H,H_0 collinear

Source: Turkey NMO 2008 Problem 1

December 2, 2008
geometrycircumcirclesymmetrygeometric transformationhomothetyEulerratio

Problem Statement

Given an acute angled triangle ABC ABC , O O is the circumcenter and H H is the orthocenter.Let A1 A_1,B1 B_1,C1 C_1 be the midpoints of the sides BC BC,AC AC and AB AB respectively. Rays [HA1 [HA_1,[HB1 [HB_1,[HC1 [HC_1 cut the circumcircle of ABC ABC at A0 A_0,B0 B_0 and C0 C_0 respectively.Prove that O O,H H and H0 H_0 are collinear if H0 H_0 is the orthocenter of A0B0C0 A_0B_0C_0
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