OIM 2004 Problem 5

Given a scalene triangle $ABC$ . Let $A'$ , $B'$ , $C'$ be the points where the internal bisectors of the angles $CAB$ , $ABC$ , $BCA$ meet the sides $BC$ , $CA$ , $AB$ , respectively. Let the line $BC$ meet the perpendicular bisector of $AA'$ at $A''$ . Let the line $CA$ meet the perpendicular bisector of $BB'$ at $B'$ . Let the line $AB$ meet the perpendicular bisector of $CC'$ at $C''$ . Prove that $A''$ , $B''$ and $C''$ are collinear.