Question 24
Let $\Omega$ be the circumcircle of triangle $ABC$, $I$ its incentre and $J$ its $A$-excentre. Let $T$ be the reflection of $J$ in $BC$, and $P$ the intersection of $BC$ and $AT$. If the circumcircle of $\triangle AIP$ meets $BC$ at $X \neq P$, and the point $Y \neq A$ on $\Omega$ is such that $\overline{IA} = \overline{YI}$, prove that the circumcircle of $\triangle IXY$ is tangent to the line $AI$.