Question 161
Let $ABC$ be an acute triangle, let $A'$, $B'$ and $C'$ be the reflections of the vertices $A$, $B$ and $C$ with respect to $BC$, $CA$ and $AB$, respectively, and let the circumcircles of triangles $ABB'$ and $ACC'$ meet again at $A_1$. Points $B_1$ and $C_1$ are defined similarly. Prove that the lines $AA_1$, $BB_1$ and $CC_1$ have a common point.