2008 IMO Shortlist G6

There is given a convex quadrilateral $A B C D$. Prove that there exists a point $P$ inside the quadrilateral such that $$\angle P A B+\angle P D C=\angle P B C+\angle P A D=\angle P C D+\angle P B A=\angle P D A+\angle P C B=90^{\circ}$$ if and only if the diagonals $A C$ and $B D$ are perpendicular.

给定一个凸四边形$A B C D$。证明四边形内部存在点 $P$ 使得 $$\angle P A B+\angle P D C=\angle P B C+\angle P A D=\angle P C D+\angle P B A=\angle P D A+\angle P C B=90^{\circ}$$ 当且仅当对角线 $A C$ 和 $B D$ 垂直。