Question 20
(IRE) Let $D$ be an internal point on the side $BC$ of a triangle $ABC$. The line $AD$ meets the circumcircle of $ABC$ again at $X$. Let $P$ and $Q$ be the feet of the perpendiculars from $X$ to $AB$ and $AC$, respectively, and let $\gamma$ be the circle with diameter $XD$. Prove that the line $PQ$ is tangent to $\gamma$ if and only if $AB=AC$.