2006 IMO Shortlist G9

Points $A_{1}, B_{1}, C_{1}$ are chosen on the sides $B C, C A, A B$ of a triangle $A B C$, respectively. The circumcircles of triangles $A B_{1} C_{1}, B C_{1} A_{1}, C A_{1} B_{1}$ intersect the circumcircle of triangle $A B C$ again at points $A_{2}, B_{2}, C_{2}$, respectively $\left(A_{2} \neq A, B_{2} \neq B, C_{2} \neq C\right)$. Points $A_{3}, B_{3}, C_{3}$ are symmetric to $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of the sides $B C, C A, A B$ respectively. Prove that the triangles $A_{2} B_{2} C_{2}$ and $A_{3} B_{3} C_{3}$ are similar. (Russia)

分别在三角形 $A B C$ 的边 $B C、C A、A B$ 上选择点 $A_{1}、B_{1}、C_{1}$。三角形 $A B_{1} C_{1}、B C_{1} A_{1}、C A_{1} B_{1}$ 的外接圆再次与三角形 $A B C$ 的外接圆相交于点 $A_{2}、B_{2}、C_{2}$，分别 \left(A_{2} \neq A, B_{2} \neq B, C_{2} \neq C\右)。点$A_{3}、B_{3}、C_{3}$ 分别相对于边$B C、C A、A B$ 的中点与$A_{1}、B_{1}、C_{1}$ 对称。证明三角形 $A_{2} B_{2} C_{2}$ 和 $A_{3} B_{3} C_{3}$ 相似。 (俄罗斯)