Question 8
Two circles $\Gamma_{1}$ and $\Gamma_{2}$ intersect at points $A$ and $Z$ (with $A \neq Z$). Let $B$ be the centre of $\Gamma_{1}$ and let $C$ be the centre of $\Gamma_{2}$. The exterior angle bisector of $\angle B A C$ intersects $\Gamma_{1}$ again at $X$ and $\Gamma_{2}$ again at $Y$. Prove that the interior angle bisector of $\angle B Z C$ passes through the circumcentre of $\triangle X Y Z$.
For points $P, Q, R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle P Q S$ is the line that divides $\angle P Q S$ into two equal angles, while the exterior angle bisector of $\angle P Q S$ is the line that divides $\angle R Q S$ into two equal angles.
For points $P, Q, R$ that lie on a line $\ell$ in that order, and a point $S$ not on $\ell$, the interior angle bisector of $\angle P Q S$ is the line that divides $\angle P Q S$ into two equal angles, while the exterior angle bisector of $\angle P Q S$ is the line that divides $\angle R Q S$ into two equal angles.