MathNet 2014 Question 42

Let $ABC$ be an acute triangle with $|AB|\ne|AC|$. Let $D,E$ be the midpoints of $AB,AC$, respectively. The circumcircles of $BCD,BCE$ meet the circumcircle of $ADE$ again at $P,Q$, respectively, where $P\ne D$ and $Q\ne E$. Prove that $|AP|=|AQ|$.

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