Problem (2018RMMp1): Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
AlphaGeometry formalization: a b c = triangle; d = on_circum a b c; p = on_line a b; q = on_line a c, on_line d p; k = on_pline p c d, on_line b c; l = on_pline q b d, on_line b c; o1 = circumcenter b k p; o2 = circumcenter c l q; t = on_circle o1 b, on_circle o2 c ? coll o1 o2 t
HAGeo formalization: A B C = triangle; (ABC) = circle A B C; D = on_circle (ABC); AB = line A B; P = on_line AB; AC = line A C; DP = line D P; Q = intersection AC DP; CD = line C D; l1 = parallel_line P CD; BC = line B C; K = intersection l1 BC; BD = line B D; l2 = parallel_line Q BD; L = intersection l2 BC; O1 = circumcenter B K P; (O1) = circle_center_point O1 B; O2 = circumcenter C L Q; (O2) = circle_center_point O2 C; T = intersection (O1) (O2); Prove: collinear O1 O2 T
Difficulty score: 2.1