# 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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