Question 182
Let $\triangle ABC$ be a triangle and let $P$ and $Q$ be points on sides $AB$ and $AC$, respectively, such that $AP = AQ$ and line $PQ$ passes through the incenter $I$ of triangle $ABC$. Let $M$ be the second intersection point of the circumcircles of triangles $BPI$ and $CQI$. The lines $PM$ and $BI$ intersect at $D$ and the lines $QM$ and $CI$ meet at $E$. Prove that the line $MI$ passes through the midpoint of segment $DE$.