Question 22
Let $A B C$ be an equilateral triangle. Points $A_{1}, B_{1}, C_{1}$ lie inside triangle $A B C$ such that triangle $A_{1} B_{1} C_{1}$ is scalene, $B A_{1}=A_{1} C, C B_{1}=B_{1} A, A C_{1}=C_{1} B$ and
$$
\angle B A_{1} C+\angle C B_{1} A+\angle A C_{1} B=480^{\circ} .
$$
Lines $B C_{1}$ and $C B_{1}$ intersect at $A_{2}$; lines $C A_{1}$ and $A C_{1}$ intersect at $B_{2}$; and lines $A B_{1}$ and $B A_{1}$ intersect at $C_{2}$.
Prove that the circumcircles of triangles $A A_{1} A_{2}, B B_{1} B_{2}, C C_{1} C_{2}$ have two common points.
$$
\angle B A_{1} C+\angle C B_{1} A+\angle A C_{1} B=480^{\circ} .
$$
Lines $B C_{1}$ and $C B_{1}$ intersect at $A_{2}$; lines $C A_{1}$ and $A C_{1}$ intersect at $B_{2}$; and lines $A B_{1}$ and $B A_{1}$ intersect at $C_{2}$.
Prove that the circumcircles of triangles $A A_{1} A_{2}, B B_{1} B_{2}, C C_{1} C_{2}$ have two common points.