Let $ABC$ be a triangle inscribed in circle $\omega$ and $P$ a point in its interior. The lines $AP,BP$ and $CP$ intersect circle $\omega$ for the second time at $D,E$ and $F,$ respectively. If $A',B',C'$ are the reflections of $A,B,C$ with respect to the lines $EF,FD,DE,$ respectively, prove that the triangles $ABC$ and $A'B'C'$ are similar.