Question 61
Given a triangle $ABC$ with $AC = BC \gt AB$, let $E$ and $F$ be the midpoints of $AC$ and $AB$, respectively. The perpendicular bisector $\ell$ of $AC$ meets $AB$ at $K$ and the line parallel to $KC$ and passing through the point $B$ intersects $AC$ at a point $L$. For a point $P$ on the line segment $BF$, let $H$ be the orthocenter of the triangle $ACP$. The line segments $BH$ and $CP$ meet at a point $J$ and the lines $FJ$ and $\ell$ meet at a point $M$. Let $W$ be the intersection point of $FL$ and $\ell$. Show that $AW = BW$ if and only if the points $B, E, F, M$ are concyclic.