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Concurent!

Source: Viet Nam TST 2009, Problem 1

April 20, 2009
geometrygeometric transformationreflectioncircumcircleEulerhomothetygeometry proposed

Problem Statement

Let an acute triangle ABC ABC with curcumcircle (O) (O). Call A1,B1,C1 A_1,B_1,C_1 are foots of perpendicular line from A,B,C A,B,C to opposite side. A2,B2,C2 A_2,B_2,C_2 are reflect points of A1,B1,C1 A_1,B_1,C_1 over midpoints of BC,CA,AB BC,CA,AB respectively. Circle (AB2C2),(BC2A2),(CA2B2) (AB_2C_2),(BC_2A_2),(CA_2B_2) cut (O) (O) at A3,B3,C3 A_3,B_3,C_3 respectively. Prove that: A1A3,B1B3,C1C3 A_1A_3,B_1B_3,C_1C_3 are concurent.
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