RIOPLATENSE 2018 Problem 4

Let $ABC$ be an acute triangle with $AC> AB$ . be $\Gamma$ the circumcircle circumscribed to the triangle $ABC$ and $D$ the midpoint of the smallest arc $BC$ of this circle. Let $E$ and $F$ points of the segments $AB$ and $AC$ respectively such that $AE = AF$ . Let $P \neq A$ be the second intersection point of the circumcircle circumscribed to $AEF$ with $\Gamma$ . Let $G$ and $H$ be the intersections of lines $PE$ and $PF$ with $\Gamma$ other than $P$ , respectively. Let $J$ and $K$ be the intersection points of lines $DG$ and $DH$ with lines $AB$ and $AC$ respectively. Show that the $JK$ line passes through the midpoint of $BC$