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Source: IMO ShortList 1991, Problem 1 (PHI 3), China 2005 TST 1

March 21, 2005
geometryTriangleorthogonalconcurrencyIMO Shortlist

Problem Statement

Given a point P P inside a triangle △ABC \triangle ABC. Let D D, E E, F F be the orthogonal projections of the point P P on the sides BC BC, CA CA, AB AB, respectively. Let the orthogonal projections of the point A A on the lines BP BP and CP CP be M M and N N, respectively. Prove that the lines ME ME, NF NF, BC BC are concurrent. Original formulation: Let ABC ABC be any triangle and P P any point in its interior. Let P1,P2 P_1, P_2 be the feet of the perpendiculars from P P to the two sides AC AC and BC. BC. Draw AP AP and BP, BP, and from C C drop perpendiculars to AP AP and BP. BP. Let Q1 Q_1 and Q2 Q_2 be the feet of these perpendiculars. Prove that the lines Q1P2,Q2P1, Q_1P_2,Q_2P_1, and AB AB are concurrent.
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