Question 135
Let $ABC$ isosceles triangle with $AB = AC$ and incenter $I$. Let $\omega$ be the circumcircle of $ABC$. The line $BI$ meets $\omega$ again at point $P$, and the line $CI$ meets $\omega$ again at point $Q$. Let $D$ be a point on the arc $BC$ of $\omega$ not containing $A$, different from $B$ and $C$. The line $BI$ meets the segment $AD$ at point $M$, and the segment $DQ$ at point $X$. The line $CI$ meets the segment $AD$ at point $N$, and the segment $DP$ at point $Y$. Prove that the lines $BN$ and $CM$ intersect on the circumcircle of triangle $XIY$.