Question 67
Let $I$ be the incenter of the triangle $ABC$, and let the incircle touch the sides $CA, AB$ at the points $E, F$, respectively. Let the reflection points of $E, F$ with respect to $I$ be the points $G, H$, respectively. Suppose that the lines $GH$ and $BC$ intersect at the point $Q$. Denote the midpoint of the side $BC$ by $M$. Prove that $IQ$ and $IM$ are perpendicular to each other.
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