2006 IMO Shortlist G2

Let $A B C D$ be a trapezoid with parallel sides $A B>C D$. Points $K$ and $L$ lie on the line segments $A B$ and $C D$, respectively, so that $A K / K B=D L / L C$. Suppose that there are points $P$ and $Q$ on the line segment $K L$ satisfying $$\angle A P B=\angle B C D \quad \text { and } \quad \angle C Q D=\angle A B C \text {. }$$ Prove that the points $P, Q, B$ and $C$ are concyclic. (Ukraine)

设 $A B C D$ 为边平行的梯形 $A B>C D$。点$K$和$L$分别位于线段$A B$和$C D$上，因此$A K / K B=D L / L C$。假设线段$K L$上有点$P$和$Q$满足$$\angle A P B=\angle B C D \quad \text { 和 } \quad \angle C Q D=\angle A B C \text { 。 }$$ 证明点$P、Q、B$ 和$C$ 是同循环的。 (乌克兰)