# Picture this ....

Calculus Level 5

A circle of radius $$r \lt 1$$ is internally tangent to a unit circle. A chord is drawn inside the unit circle that is tangent to the radius $$r$$ circle and perpendicular to the line joining the centers of the two circles.

Let $$O$$ be the center of the smaller circle, and let $$P, Q$$ be the endpoints of the chord as described above. The value of $$r$$ that maximizes the perimeter of triangle $$OPQ$$ can be written as $$\dfrac{a - \sqrt{b}}{c}$$, where $$a, b, c$$ are positive coprime integers and $$b$$ is square-free. Find $$a + b + c$$.

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