ÖMO 1985 Problem 3

A line meets the lines containing sides $BC,CA,AB$ of a triangle $ABC$ at $A_1,B_1,C_1,$ respectively. Points $A_2,B_2,C_2$ are symmetric to $A_1,B_1,C_1$ with respect to the midpoints of $BC,CA,AB,$ respectively. Prove that $A_2,B_2,$ and $C_2$ are collinear.