Question 22
Points $A_{1}$, $B_{1}$, $C_{1}$ are chosen on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively. The circumcircles of triangles $AB_{1}C_{1}$, $BC_{1}A_{1}$, $CA_{1}B_{1}$ intersect the circumcircle of triangle $ABC$ again at points $A_{2}$, $B_{2}$, $C_{2}$, respectively ($A_{2} \neq A$, $B_{2} \neq B$, $C_{2} \neq C$). Points $A_{3}$, $B_{3}$, $C_{3}$ are symmetric to $A_{1}$, $B_{1}$, $C_{1}$ with respect to the midpoints of the sides $BC$, $CA$, $AB$ respectively. Prove that the triangles $A_{2}B_{2}C_{2}$ and $A_{3}B_{3}C_{3}$ are similar.