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a concurrency problem

Source: I.F.Sharygin contest 2009 - Correspondence round - Problem 17

May 31, 2009
geometrycircumcircleratiogeometry proposed

Problem Statement

Given triangle ABC ABC and two points X X, Y Y not lying on its circumcircle. Let A1 A_1, B1 B_1, C1 C_1 be the projections of X X to BC BC, CA CA, AB AB, and A2 A_2, B2 B_2, C2 C_2 be the projections of Y Y. Prove that the perpendiculars from A1 A_1, B1 B_1, C1 C_1 to B2C2 B_2C_2, C2A2 C_2A_2, A2B2 A_2B_2, respectively, concur if and only if line XY XY passes through the circumcenter of ABC ABC.
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