Problem (2015AsiaPacificMOp1): Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively. Prove that $AB = V W$
AlphaGeometry formalization: a b c = triangle; d = on_line b c; x = on_line a b; y = on_line a c, on_line d x; z = on_circum b x d, on_circum a b c; v = on_line z d, on_circum a b c; w = on_line z y, on_circum a b c ? cong a b v w
HAGeo formalization: A B C = triangle; BC = line B C; D = on_line BC; l = line_through D; AB = line A B; X = intersection l AB; AC = line A C; Y = intersection l AC; (ABC) = circumcircle A B C; (BDX) = circle B D X; Z = intersection (BDX) (ABC); DZ = line D Z; V = intersection DZ (ABC); YZ = line Y Z; W = intersection YZ (ABC); VW = line V W; Prove: cong A B V W
Difficulty score: 1.7