# Twin Circles

Geometry Level 5

Triangle ABC is inscribed in a circle $$\Gamma_1$$ so that BC is a diameter. Circle $$\Gamma_2$$ is inscribed in ABC, and circle $$\Gamma_3$$ is tangent to both segment AB and minor arc AB so that the two points of tangency lie on the same diameter of $$\Gamma_1$$. If $$\Gamma_2$$ and $$\Gamma_3$$ are congruent, then the extended ratio BC : AC : AB can be written a : b : c, where a, b, and c are positive coprime integers. Find $$a+b+c$$.

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