Problem (2019BalkanMOp3): Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that: $1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively. $2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively. Prove that $KL$ and $ST$ intersect on the line $BC$.
AlphaGeometry formalization: a b c = triangle; x = on_line b c; y = on_line b c, on_aline y a b c a x; k = foot b a x; s = foot b a y; t = foot c a x; l = foot c a y; p = on_line k l, on_line s t ? coll p b c
HAGeo formalization: A B C = acute_triangle; BC = line B C; X = on_line BC; l1 = angle_equal1 A B C A X; Y = intersection BC l1; AX = line A X; K = foot B AX; AY = line A Y; S = foot B AY; T = foot C AX; L = foot C AY; KL = line K L; ST = line S T; P = intersection KL ST; Prove: collinear P B C
Difficulty score: 3.1