Problem (2022BalkanMOp1): Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$. Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$. Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$. The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$. Prove that the line $YZ$ passes through the midpoint of the line segment $AC$.
AlphaGeometry formalization: a b c = triangle; o = circumcenter a b c; x = on_tline a o a, on_tline b o b; y = foot o c x; z = on_line a x, on_pline c a b; m = midpoint a c ? coll y z m
HAGeo formalization: A B C = acute_triangle; O = circumcenter A B C; (O) = circle_center_point O A; OA = line O A; OB = line O B; ta = perpendicular_line A OA; tb = perpendicular_line B OB; X = intersection ta tb; CX = line C X; Y = foot O CX; AB = line A B; l1 = parallel_line C AB; Z = intersection l1 ta; M = midpoint A C; Prove: collinear Y Z M
Difficulty score: 2.0