Question 156
Let $ABC$ be an acute triangle with $AB \lt AC$, and let $\Gamma$ be the circumcircle of $ABC$. Points $X$ and $Y$ lie on $\Gamma$ so that $XY$ and $BC$ intersect on the external angle bisector of $\angle BAC$. Suppose that the tangents to $\Gamma$ at $X$ and $Y$ intersect at a point $T$ on the same side of $BC$ as $A$, and that $TX$ and $TY$ intersect $BC$ at $U$ and $V$, respectively. Let $J$ be the centre of the excircle of triangle $TUV$ opposite the vertex $T$.
Prove that $AJ$ bisects $\angle BAC$.
Prove that $AJ$ bisects $\angle BAC$.