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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. By solving this ...
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
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 ...
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.
An osculating circle Osculating circles of the Archimedean spiral, nested by the Tait–Kneser theorem. "The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." [1] An osculating circle is a circle that best approximates the curvature of a curve at a specific point.
Descartes' theorem; E. Eyeball theorem; F. Five circles theorem; I. Inscribed angle; ... Six circles theorem; T. Tangent–secant theorem This page was ...
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.
He rediscovered the Descartes' theorem in 1936 and published it as a poem, "The Kiss Precise", [28] quoted at Problem of Apollonius. The kissing circles in this problem are sometimes known as Soddy circles .