A special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles (also called the kissing circles problem). There are two solutions: a small circle surrounded by the three original circles, and a large circle surrounding the original three. Frederick Soddy gave the formula for finding the radius of the so-called inner and outer Soddy circles given the radii of the other three. The relationship is 2(κ_1^2 + κ_2^2 + κ_3^2 + κ_4^2) = (κ_1 + κ_2 + κ_3 + κ_4)^2, where κ_i = 1/r_i are the curvatures of the circles with radii r_i. Here, the negative solution corresponds to the outer Soddy circle and the positive solution to the inner Soddy circle.
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