1994 AIME Problem 15

Given a point P on a triangular piece of paper A B C, 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 A B C if these creases, which number three unless P is one of the vertices, do not intersect. Suppose that A B=36, A C=72, and \angle B=90^{\circ}. Then the area of the set of all fold points of \triangle A B C 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?