Playing with Right-Angled Triangles!

Geometry Level 4

Let \(ABC\) be a right-angled triangle with hypotenuse \(AB\) and altitude \(CF\), where \(F\) lies on \(AB\). The circle through \(F\) centred at \(B\) and another circle of the same radius centred at \(A\) intersect on the side \(CB\). If the ratio between the lengths of \(FB\) and \(BC\) can be expressed as \(\left( \dfrac{A}{B} \right)^{C/D}\), for positive integers \(A,B,C,D\), find the minimum value of \(A+B+C+D\).

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