APMO 2016 Problem 1

We say that a triangle $A B C$ is great if the following holds: for any point $D$ on the side $B C$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $A B$ and $A C$, respectively, then the reflection of $D$ in the line $P Q$ lies on the circumcircle of the triangle $A B C$.

Prove that triangle $A B C$ is great if and only if $\angle A=90^{\circ}$ and $A B=A C$.