Question 39
Let $ABC$ be an acute-angled triangle with $AC \gt AB$. Let $\Gamma$ be the circumference circumscribed about the triangle $ABC$ and $D$ the midpoint of the smaller arc $BC$ of $\Gamma$. Let $E$ and $F$ be points in the segments $AB$ and $AC$ respectively such that $AE = AF$. Let $P \ne A$ be the second intersection point of the circumference circumscribed about the triangle $AEF$ with $\Gamma$. Let $G$ and $H$ be the points, different from $P$, where the lines $PE$ and $PF$ intersect $\Gamma$, respectively. Let $J$ and $K$ be the intersections of the lines $DG$ and $DH$ with the lines $AB$ and $AC$ respectively. Prove that the line $JK$ passes through the midpoint of $BC$.