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  2. Descartes' theorem - Wikipedia

    en.wikipedia.org/wiki/Descartes'_theorem

    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.

  3. Tangent circles - Wikipedia

    en.wikipedia.org/wiki/Tangent_circles

    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

  4. Sum of squares - Wikipedia

    en.wikipedia.org/wiki/Sum_of_squares

    The British flag theorem for rectangles equates two sums of two squares; The parallelogram law equates the sum of the squares of the four sides to the sum of the squares of the diagonals; Descartes' theorem for four kissing circles involves sums of squares; The sum of the squares of the edges of a rectangular cuboid equals the square of any ...

  5. Problem of Apollonius - Wikipedia

    en.wikipedia.org/wiki/Problem_of_Apollonius

    Descartes' theorem was rediscovered independently in 1826 by Jakob Steiner, [50] in 1842 by Philip Beecroft, [2] [49] and again in 1936 by Frederick Soddy. [51] Soddy published his findings in the scientific journal Nature as a poem, The Kiss Precise, of which the first two stanzas are reproduced below. The first stanza describes Soddy's ...

  6. Apollonian gasket - Wikipedia

    en.wikipedia.org/wiki/Apollonian_gasket

    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.

  7. Category:Theorems about circles - Wikipedia

    en.wikipedia.org/.../Category:Theorems_about_circles

    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 ...

  8. Soddy circles of a triangle - Wikipedia

    en.wikipedia.org/wiki/Soddy_circles_of_a_triangle

    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.

  9. Osculating circle - Wikipedia

    en.wikipedia.org/wiki/Osculating_circle

    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.