Question 3
Let $ABC$ be an equilateral triangle. Points $A_1$, $B_1$, $C_1$ lie inside triangle $ABC$ such that triangle $A_1B_1C_1$ is scalene, $BA_1 = A_1C$, $CB_1 = B_1A$, $AC_1 = C_1B$ and
$$
\angle BA_1C + \angle CB_1A + \angle AC_1B = 480^\circ.
$$
Lines $BC_1$ and $CB_1$ intersect at $A_2$; lines $CA_1$ and $AC_1$ intersect at $B_2$; and lines $AB_1$ and $BA_1$ intersect at $C_2$.
Prove that the circumcircles of triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ have two common points.
$$
\angle BA_1C + \angle CB_1A + \angle AC_1B = 480^\circ.
$$
Lines $BC_1$ and $CB_1$ intersect at $A_2$; lines $CA_1$ and $AC_1$ intersect at $B_2$; and lines $AB_1$ and $BA_1$ intersect at $C_2$.
Prove that the circumcircles of triangles $AA_1A_2$, $BB_1B_2$, $CC_1C_2$ have two common points.