SHARYGIN 2007 Problem 16

On two sides of an angle, points $A, B$ are chosen. The midpoint $M$ of the segment $AB$ belongs to two lines such that one of them meets the sides of the angle at points $A_1, B_1$ , and the other at points $A_2, B_2$ . The lines $A_1B_2$ and $A_2B_1$ meet $AB$ at points $P$ and $Q$ . Prove that $M$ is the midpoint of $PQ$ .