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intersection of (BPK), (DQK) lies on diagonal BD of trapezoid ABCD

Source: VI Soros Olympiad 1990-00 R1 9.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

May 27, 2024
geometrytrapezoidconcurrencyConcyclic

Problem Statement

A straight line is drawn through an arbitrary internal point KK of the trapezoid ABCDABCD, intersecting the bases of BCBC and ADAD at points PP and QQ, respectively. The circles circumscribed around the triangles BPKBPK and DQKDQK intersect, besides the point KK, also at the point LL. Prove that the point LL lies on the diagonal BDBD.