2006 IMO Shortlist G8

Let $A B C D$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that $$\angle P A B+\angle P D C \leq 90^{\circ} \quad \text { and } \quad \angle P B A+\angle P C D \leq 90^{\circ} \text {. }$$ Prove that $A B+C D \geq B C+A D$. (Poland)

令 $A B C D$ 为凸四边形。通过点 $A$ 和 $D$ 的圆以及通过点 $B$ 和 $C$ 的圆在四边形内部的点 $P$ 处外切。假设 \angle P A B+\angle P D C \leq 90^{\circ} \quad \text { 和 } \quad \angle P B A+\angle P C D \leq 90^{\circ} \text {。证明$A B+C D \geq B C+A D$。 (波兰)