KöMaL 2007 B4025

A regular triangle is drawn on the side $BC$ of a triangle ABC towards the outside, and one on the side $CA$ towards the inside. The third vertices of the regular triangles are $A^*$ and $B^*$, respectively. The reflection of vertex $C$ in the line $AB$ is $C$'. Prove that the points $A^*$, $B^*$ és $C$' are collinear.