Question 15
Given a point $P$ on a triangular piece of paper $ABC,$ consider the creases that are formed in the paper when $A,$ $B,$ and $C$ are folded onto $P.$ Let us call $P$ a fold point of $\triangle ABC$ if these creases, which number three unless $P$ is one of the vertices, do not intersect. Suppose that $AB=36,$ $AC=72,$ and $\angle B=90^\circ.$ Then the area of the set of all fold points of $\triangle ABC$ can be written in the form $q\pi-r\sqrt s,$ where $q,$ $r,$ and $s$ are positive integers and $s$ is not divisible by the square of any prime. What is $q+r+s?$