ToT 1998-fall

In a triangle $ABC$ the points $A'$ , $B'$ and $C'$ lie on the sides $BC$ , $CA$ and $AB$ , respectively. It is known that $\angle AC'B' = \angle B'A'C$ , $\angle CB'A' = \angle A'C'B$ and $\angle BA'C' = \angle C'B'A$ . Prove that $A'$ , $B'$ and $C'$ are the midpoints of the corresponding sides.