# Radius of a Circle, Given Tangent

Geometry Level 5

$$P$$ is a point outside of circle $$\Gamma$$. The tangent from $$P$$ to $$\Gamma$$ touches at $$A$$. A line from $$P$$ intersects $$\Gamma$$ at $$B$$ and $$C$$ such that $$\angle ACP = 120^\circ$$. If $$AC = 16$$ and $$AP = 19$$, then the radius of $$\Gamma$$ can be expressed as $$\frac {a \sqrt{b} }{c}$$, where $$b$$ is an integer not divisible by the square of any prime and $$a, c$$ are coprime positive integers. What is $$a + b + c$$?

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