# Bouncing off of the Sides of A Triangle Forever

Geometry Level 5

Let $$ABC$$ be a triangle such that $$BC=5$$, $$CA=8$$, and $$AB=9$$. A ball is launched from a point $$X$$ on segment $$BC$$ such that it first bounces off of segment $$CA$$ at a point $$Y$$, then bounces off of segment $$AB$$ at a point $$Z$$, then bounces off of segment $$BC$$ at $$X$$, then bounces off of segment $$CA$$ at $$Y$$, and so on, traversing the perimeter of triangle $$XYZ$$ forever.

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

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