HMMT 2007 T12

Let $A B C D$ be a cyclic quadrilateral, and let $P$ be the intersection of its two diagonals. Points $R, S, T$, and $U$ are feet of the perpendiculars from $P$ to sides $A B, B C, C D$, and $A D$, respectively. Show that quadrilateral $R S T U$ is bicentric if and only if $A C \perp B D$. (Note that a quadrilateral is called inscriptible if it has an incircle; a quadrilateral is called bicentric if it is both cyclic and inscriptible.)