This triangle is arbitrary so how do we determine the circumradius?

Geometry Level 5

In \(\Delta ABC\), circle \( \Gamma_1 \) passes through \(A\) and \(B\), and is tangent to \( BC \). Circle \( \Gamma_2 \) passes through \(A\) and \(C\), and is tangent to \( BC \). Let the radii of these two circles be 12 and 15 units respectively. If the circumradius of \(\Delta ABC\) can be represented as \(a \sqrt{b}+c\), where the surd has been simplified completely and \(a\), \(b\) and \(c\) are integers, find \(a+b+c\).

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