Problem (2019CGMOp1): Let $ABCD$ be a cyclic quadrilateral with circumcircle $\odot O.$ The lines tangent to $\odot O$ at $A,B$ intersect at $L.$ $M$ is the midpoint of the segment $AB.$ The line passing through $D$ and parallel to $CM$ intersects $ \odot (CDL) $ at $F.$ Line $CF$ intersects $DM$ at $K,$ and intersects $\odot O$ at $E$ (different from point $C$). Prove that $EK=DK.$
AlphaGeometry formalization: a b c = triangle; d = on_circum a b c; o = circumcenter a b c; l = on_tline a o a, on_tline b o b; m = midpoint a b; o1 = circumcenter c d l; f = on_pline d c m, on_circle o1 c; k = on_line c f, on_line d m; e = on_line c f, on_circle o c ? cong e k d k
HAGeo formalization: A B C = triangle; O = circumcenter A B C; (O) = circle_center_point O A; D = on_circle (O); OA = line O A; OB = line O B; l1 = perpendicular_line A OA; l2 = perpendicular_line B OB; L = intersection l1 l2; M = midpoint A B; CM = line C M; l3 = parallel_line D CM; O1 = circumcenter C D L; (O1) = circle_center_point O1 C; F = intersection l3 (O1); CF = line C F; DM = line D M; K = intersection CF DM; E = intersection CF (O); Prove: cong E K D K
Difficulty score: 2.9