Sitting pretty

Geometry Level 3

Suppose three (thin) metal rods of length \(6, 7\) and \(8\), are connected to form a triangle. A solid sphere of radius \(4\) is then positioned to "sit" in the triangle so that it is tangent to each of the three rods. Let \(H\) be the height of the top of the sphere above the plane of the triangle. If \(H = \dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers, then find \(a + b\).

(Assume that the rods are of negligible thickness.)

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