Question 62
Let $ABCD$ 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 $AB$, $BC$, $CD$, and $AD$, respectively. Show that quadrilateral $RSTU$ is bicentric if and only if $AC \perp BD$. (Note that a quadrilateral is called inscriptible if it has an incircle; a quadrilateral is called bicentric if it is both cyclic and inscriptible.)
