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Geometry Level 3

There are two circles, one with center \(A\) and radius 1, the other with center \(B\) and radius 2. The distance \(AB\) is 6. A third circle of unknown radius is tangent to both of these circles and there exists a straight line which

(1) is tangent to all three circles, and
(2) intersects the segment \(AB\).

Find the radius of the third circle.

There are two solutions, \(R\) and \(r\). Evaluate \( R - r \).


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