Question 121
Let $\gamma_1$ and $\gamma_2$ be external circles in the plane, centred at $O_1$ and $O_2$, respectively. One of their external tangents touches $\gamma_1$ at $A_1$ and $\gamma_2$ at $A_2$. One of their internal tangents touches $\gamma_1$ at $B_1$ and $\gamma_2$ at $B_2$, and the other touches $\gamma_1$ at $C_1$ and $\gamma_2$ at $C_2$. Let $B_1B_2$ and $C_1C_2$ cross at $O$. Let $X$ be the point where $A_2O$ crosses $\gamma_1$ and $OX \lt OB_1$, and let $Y$ be the point where $A_1O$ crosses $\gamma_2$ and $OY \lt OB_2$. The perpendicular at $X$ to $OX$ crosses the line $O_1B_1$ at $P$, and the perpendicular at $Y$ to $OY$ crosses the line $O_2C_2$ at $Q$. Prove that $PQ$ and $A_1A_2$ are parallel.
