BG-MO 2000 Problem 2

Let be given an acute triangle $ABC$ . Show that there exist unique points $A_1 \in BC$ , $B_1 \in CA$ , $C_1 \in AB$ such that each of these three points is the midpoint of the segment whose endpoints are the orthogonal projections of the other two points on the corresponding side. Prove that the triangle $A_1B_1C_1$ is similar to the triangle whose side lengths are the medians of $\triangle ABC$ .