The Descartes circle theorem, also called the kissing circles problem, is a special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles. 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.
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