OIM 2015 Problem 4

Let $ABC$ be an acute triangle and let $D$ be the foot of the perpendicular from $A$ to side $BC$ . Let $P$ be a point on segment $AD$ . Lines $BP$ and $CP$ intersect sides $AC$ and $AB$ at $E$ and $F$ , respectively. Let $J$ and $K$ be the feet of the peroendiculars from $E$ and $F$ to $AD$ , respectively. Show that $\frac{FK}{KD}=\frac{EJ}{JD}$ .