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Kissing circles. Given three mutually tangent circles (black), there are, in general, two possible answers (red) as to what radius a fourth tangent circle can have.In geometry, Descartes' theorem states that for every four kissing, or mutually tangent circles, the radii of the circles satisfy a certain quadratic equation.
Tangent lines to circles; Circle packing theorem, the result that every planar graph may be realized by a system of tangent circles; Hexafoil, the shape formed by a ring of six tangent circles; Feuerbach's theorem on the tangency of the nine-point circle of a triangle with its incircle and excircles; Descartes' theorem; Ford circle; Bankoff circle
The size of each new circle is determined by Descartes' theorem, which states that, for any four mutually tangent circles, the radii of the circles obeys the equation (+ + +) = (+ + +). This equation may have a solution with a negative radius; this means that one of the circles (the one with negative radius) surrounds the other three.
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They are all named for Frederick Soddy, who rediscovered Descartes' theorem on the radii of mutually tangent quadruples of circles. Any triangle has three externally tangent circles centered at its vertices. Two more circles, its Soddy circles, are tangent to the three circles centered at the vertices; their centers are called Soddy centers.
René Descartes gave a formula relating the radii of the solution circles and the given circles, now known as Descartes' theorem. Solving Apollonius' problem iteratively in this case leads to the Apollonian gasket , which is one of the earliest fractals to be described in print, and is important in number theory via Ford circles and the Hardy ...
Pages in category "Theorems about circles" The following 21 pages are in this category, out of 21 total. ... Descartes' theorem; E. Eyeball theorem; F. Five circles ...
The purpose of including it was to (a) demonstrate how the generalization works in a specific case, and (b) show how the two "soddy circles" (related by the ± cases of the find-the-fourth-curvature version of Descartes' theorem) are strictly related to each-other as well as related to the other three circles, while (c) also showing how ...