3 circles \(A,B,C\) with radius \(a,b,c\) respectively, such that circles \(A,B,C\) are mutually tangent to each other. Let \(R\) be the circle that is internally tangent to circle \(A,B,C\), and \(R\) has a radius \(r\). Prove that

\[r = \frac{abc}{ab+bc+ca+2\sqrt{abc(a+b+c)}}\]

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