ISL 1991 Problem 1

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.