VMO 2008 Problem 7

Let $AD$ is centroid of $ABC$ triangle. Let $(d)$ is the perpendicular line with $AD$ . Let $M$ is a point on $(d)$ . Let $E, F$ are midpoints of $MB, MC$ respectively. The line through point $E$ and perpendicular with $(d)$ meet $AB$ at $P$ . The line through point $F$ and perpendicular with $(d)$ meet $AC$ at $Q$ . Let $(d')$ is a line through point $M$ and perpendicular with $PQ$ . Prove $(d')$ always pass a fixed point.

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