# Circle within a Circle

Geometry Level 3

Equilateral triangle $$ABC$$ has a circumcircle $$\Gamma$$ with center $$O$$ and circumradius $$10$$. Another circle $$\Gamma_1$$ is drawn inside $$\Gamma$$ such that it is tangential to radii $$OC$$ and $$OB$$ and circle $$\Gamma$$. The radius of $$\Gamma_1$$ can be expressed in the form $$a \sqrt{b} -c$$, where $$a, b$$ and $$c$$ are positive integers, and $$b$$ is not divisible by the square of any prime. What is the value of $$a + b + c$$?

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