Problem (2017BalkanMOp2): Consider an acute-angled triangle $ABC$ with $AB<AC$ and let $\omega$ be its circumscribed circle. Let $t_B$ and $t_C$ be the tangents to the circle $\omega$ at points $B$ and $C$, respectively, and let $L$ be their intersection. The straight line passing through the point $B$ and parallel to $AC$ intersects $t_C$ in point $D$. The straight line passing through the point $C$ and parallel to $AB$ intersects $t_B$ in point $E$. The circumcircle of the triangle $BDC$ intersects $AC$ in $T$, where $T$ is located between $A$ and $C$. The circumcircle of the triangle $BEC$ intersects the line $AB$ (or its extension) in $S$, where $B$ is located between $S$ and $A$. Prove that $ST$, $AL$, and $BC$ are concurrent.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; l = on_tline b o b, on_tline c o c; d = on_pline b a c, on_tline c o c; e = on_pline c a b, on_tline b o b; o1 = circumcenter b d c; t = on_line a c, on_circle o1 b; o2 = circumcenter b e c; s = on_line a b, on_circle o2 b; x = on_line s t, on_line a l ? coll x b c
HAGeo formalization: A B C = acute_triangle; O = circumcenter A B C; (O) = circle_center_point O A; OB = line O B; OC = line O C; t1 = perpendicular_line B OB; t2 = perpendicular_line C OC; L = intersection t1 t2; AC = line A C; l1 = parallel_line B AC; D = intersection l1 t2; AB = line A B; l2 = parallel_line C AB; E = intersection l2 t1; O1 = circumcenter B C D; (O1) = circle_center_point O1 B; T = intersection AC (O1); O2 = circumcenter B C E; (O2) = circle_center_point O2 B; S = intersection AB (O2); ST = line S T; AL = line A L; X = intersection ST AL; Prove: collinear X B C
Difficulty score: 2.9