Question 20
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_{1}, R_{2}, R_{3}, R_{4}$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_{1}=31$ and $R_{2}=24$ and $R_{3}=12$, find $R_{4}$.