1994 AIME Problem 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\,?

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.