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