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Descartes' circle theorem (a.k.a. the kissing circle theorem) provides a quadratic equation satisfied by the radii of four mutually tangent circles. By solving this equation, one can determine the possible values for the radius of a fourth circle tangent to three given, mutually tangent circles.
15 Απρ 2019 · The problem of the “kissing circles” and Descartes’s circle theorem are as current today as they were some four hundred years ago. To give but one example, Descartes’s formula plays an important role in the theory of circle packings in the plane.
Alden Bradford. September 2022. Abstract. How was this proof overlooked for 181 years? We give a simple proof of Descartes’s circle theorem using Cayley-Menger determinants. 1 Introduction. Descartes’s Circle Theorem states that the radii of four mutually tangent circles r1, r2, r3, and r4 satisfy. 1. . +. r1. 1. . +. r2. 1 1. r3 r4. 2 1.
24 Ιαν 2022 · Descartes circle theorem Theorem (Descartes circle theorem, 1643) If b 1;b 2;b 3;b 4 are bends of four mutually tangent circles, then (b 1 + b 2 + b 3 + b 4) 2 = 2(b2 1 + b 2 2 + b 2 3 + b 2 4): Example 11 21 24 28 b 1 = 11, 2 = 21, 3 = 24, 4 = 28 ( 11 + 21 + 24 + 28)2 = 622 = 3844 2(( 11)2 + 212 + 242 + 282) = 2(1922) = 3844 Edna Jones The ...
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 Cartesian circle (also known as Arnauld 's circle[1]) is an example of fallacious circular reasoning attributed to French philosopher René Descartes. He argued that the existence of God is proven by reliable perception, which is itself guaranteed by God.
A Short History of Descartes’s Circle Theorem. Descartes’s circle theorem was first described by Descartes in 1643 in his correspondence with Princess Elisabeth of Bohemia, one of his pupils [5]. In a letter to her, Descartes posed the following problem [4]: which is Apollonius’s problem.
