JBMO-SL 2013 G4

Let $I$ be the incenter and $A B$ the shortest side of a triangle $A B C$. The circle with center $I$ and passing through $C$ intersects the ray $A B$ at the point $P$ and the ray $B A$ at the point $Q$. Let $D$ be the point where the excircle of the triangle $A B C$ belonging to angle $A$ touches the side $B C$, and let $E$ be the symmetric of the point $C$ with respect to $D$. Show that the lines $P E$ and $C Q$ are perpendicular.