Descartes' theorem

In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643. Frederick Soddy's 1936 poem The Kiss Precise summarizes the theorem in terms of the bends (inverse radii) of the four circles: The sum of the squares of all four bends Is half the square of their sum Special cases of the theorem apply when one or two of the circles is replaced by a straight line (with zero bend) or when the bends are integers or square numbers. A version of the theorem using complex numbers allows the centers of the circles, and not just their radii, to be calculated. With an appropriate definition of curvature, the theorem also applies in spherical geometry and hyperbolic geometry. In higher dimensions, an analogous quadratic equation applies to systems of pairwise tangent spheres or hyperspheres. Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the third century BC, Apollonius of Perga devoted an entire book to the topic, Ἐπαφαί [Tangencies]. It has been lost, and is known largely through a description of its contents by Pappus of Alexandria and through fragmentary references to it in medieval Islamic mathematics. However, Greek geometry was largely focused on straightedge and compass construction. For instance, the problem of Apollonius, closely related to Descartes' theorem, asks for the construction of a circle tangent to three given circles which need not themselves be tangent. Instead, Descartes' theorem is formulated using algebraic relations between numbers describing geometric forms. This is characteristic of analytic geometry, a field pioneered by René Descartes and Pierre de Fermat in the first half of the 17th century. Descartes discussed the tangent circle problem briefly in 1643, in two letters to Princess Elisabeth of the Palatinate.

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