HMMT 2023 T9

Let $A B C$ be a triangle with $A B<A C$. The incircle of triangle $A B C$ is tangent to side $B C$ at $D$ and intersects the perpendicular bisector of segment $B C$ at distinct points $X$ and $Y$. Lines $A X$ and $A Y$ intersect line $B C$ at $P$ and $Q$, respectively. Prove that, if $D P \cdot D Q=(A C-A B)^{2}$, then $A B+A C=3 B C$.