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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