VWO 2015 Problem 2

Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$ . Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$ , and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$ . Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$ .