Question 130
Let $O_1O_2O_3$ be an acute-angled triangle. Let $\omega_1, \omega_2, \omega_3$ be the circles with centres $O_1, O_2, O_3$ respectively such that any two of them are tangent to each other. The circumcircle of $O_1O_2O_3$ intersects $\omega_1$ at $A_1$ and $B_1$, $\omega_2$ at $A_2$ and $B_2$, and $\omega_3$ at $A_3$ and $B_3$. Prove that the incenter of the triangle determined by the lines $A_1B_1, A_2B_2, A_3B_3$ and the incenter of the triangle $O_1O_2O_3$ coincide.