Question 39
The in-circle $\Gamma$ of a triangle $ABC$ touches the side $BC$ at $D$. Let $D'$ be the point which is diametrically opposite to $D$ on the circle $\Gamma$. The tangent through $D'$ to $\Gamma$ meets $AD$ in $X$. The tangent to $\Gamma$ through $X$, other than $XD'$, touches $\Gamma$ at $N$. Prove that the circum-circle of triangle $BCN$ touches $\Gamma$ at $N$.