Question 46
Let $\Gamma_{1}$ and $\Gamma_{2}$ be two circles of unequal radii, with centres $O_{1}$ and $O_{2}$ respectively, in the plane intersecting in two distinct points $A$ and $B$. Assume that the centre of each of the circles $\Gamma_{1}$ and $\Gamma_{2}$ is outside the other. The tangent to $\Gamma_{1}$ at $B$ intersects $\Gamma_{2}$ again in $C$, different from $B$; the tangent to $\Gamma_{2}$ at $B$ intersects $\Gamma_{1}$ again in $D$, different from $B$. The bisectors of $\angle D A B$ and $\angle C A B$ meet $\Gamma_{1}$ and $\Gamma_{2}$ again in $X$ and $Y$, respectively, different from $A$. Let $P$ and $Q$ be the circumcentres of triangles $A C D$ and $X A Y$, respectively. Prove that $P Q$ is the perpendicular bisector of the line segment $O_{1} O_{2}$.