BALKAN 2019 Problem 3

Let $A B C$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $B C$ such that $\angle C A X=\angle Y A B$. Suppose that:

1. $K$ and $S$ are the feet of perpendiculars from $B$ to the lines $A X$ and $A Y$ respectively;
2. $T$ and $L$ are the feet of perpendiculars from $C$ to the lines $A X$ and $A Y$ respectively. Prove that $K L$ and $S T$ intersect on the line $B C$.