2023 IMO Shortlist G8

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. (U.S.A.)

设$A B C$ 为等边三角形。点 $A_{1}、B_{1}、C_{1}$ 位于三角形 $A B C$ 内，使得三角形 $A_{1} B_{1} C_{1}$ 为不等边三角形，$B A_{1}=A_{1} C、C B_{1}=B_{1} A、A C_{1}=C_{1} B$ 和 $$\angle B A_{1} C+\angle C B_{1} A+\角度 A C_{1} B=480^{\circ} 。$$线$BC_{1}$和$C B_{1}$相交于$A_{2}$；线 $C A_{1}$ 和 $A C_{1}$ 相交于 $B_{2}$；并且线 $A B_{1}$ 和 $B A_{1}$ 相交于 $C_{2}$。证明三角形$A A_{1} A_{2}、B B_{1} B_{2}、C C_{1} C_{2}$的外接圆有两个公共点。 (美国。)