CZ-SK-MO 2019 Problem 4

Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$ . Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$ . Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$ , respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$ . Prove that $AF\perp FJ$ .