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Given a point P inside a triangle \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 P_1, P_2 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 Q_1 and Q_2 be the feet of these perpendiculars. Prove that the lines Q_1P_2,Q_2P_1, and AB are concurrent.

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