Question 171
Let $ABC$ be a triangle, $D, E, F$ the points of tangency of the incircle with the sides $(BC)$, $(AC)$ and $(AB)$, $M$ the midpoint of $[BC]$, and $I$ the incentre of $ABC$. $G$ and $H$ are defined as the reflections of $E$ and $F$ in $I$. Let $Q$ be the intersection of the lines $(BC)$ and $(GH)$. Show that the lines $(IQ)$ and $(IM)$ are perpendicular.