Bouncing off of the Sides of A Triangle Forever

Geometry Level 5

Let ABCABC be a triangle such that BC=5BC=5, CA=8CA=8, and AB=9AB=9. A ball is launched from a point XX on segment BCBC such that it first bounces off of segment CACA at a point YY, then bounces off of segment ABAB at a point ZZ, then bounces off of segment BCBC at XX, then bounces off of segment CACA at YY, and so on, traversing the perimeter of triangle XYZXYZ forever.

The area of triangle XYZXYZ can be written in the form pqr\tfrac{p\sqrt{q}}{r}, where pp and rr are coprime positive integers and qq is a squarefree integer. Find p+q+rp+q+r.

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