Question 155
Let $ABC$ be a triangle with $AB \lt AC$. The incircle of triangle $ABC$ is tangent to side $BC$ at $D$ and intersects the perpendicular bisector of segment $BC$ at distinct points $X$ and $Y$. Lines $AX$ and $AY$ intersect line $BC$ at $P$ and $Q$, respectively. Prove that, if $DP \cdot DQ = (AC - AB)^2$, then $AB + AC = 3 BC$.