Construct a Circle

Geometry Level 4

It is well known that three noncollinear points in a plane uniquely determine a circle. Let \( (u,v) \) be the center of the circle containing the three points \(A = (1,2) \), \(B = (5,8) \), and \(C = (10,7) \). Then \(u+v\) can be written as \( \frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. Find \(a+b\).

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