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Problem 7. Let $ABC$ be an acute angle triangle with $AB < AC$. Let $M$ be the midpoint of segment $BC$. Let $E$ and $F$ be points on segments $AC$ and $AB$, respectively, such that the circumcircle of triangle $MEF$ is tangent to $BC$. The circumcircles of triangles $AEF$ and $ABC$ intersect at a point $P \neq A$. Let $Q$ be a point on the circumcircle of triangle $AEF$ such that $AQ$ is perpendicular to $BC$.
Prove that $PQ$ passes through the circumcenter of triangle $MEF$.
Solution 1Solution 2Solution 3Solution 4
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