CN-TST 2022 Problem 20

Let $ABCD$ be a convex quadrilateral, the incenters of $\triangle ABC$ and $\triangle ADC$ are $I,J$ , respectively. It is known that $AC,BD,IJ$ concurrent at a point $P$ . The line perpendicular to $BD$ through $P$ intersects with the outer angle bisector of $\angle BAD$ and the outer angle bisector $\angle BCD$ at $E,F$ , respectively. Show that $PE=PF$ .