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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.
16 Μαΐ 2014 · One example is as follows: Given any 2 -coloring on K6, then we are guaranteed to find a monochromatic subgraph of size 3. This has an interesting real-life interpretation: If we invite 6 people to a party, then at least 3 of them must be mutual acquaintances, or at least 3 of them must be mutual strangers. Share.
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 ...
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.
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. = 2 . +. r2 + . r2. 1. . +. r2.
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.
30 Νοε 2022 · Descartes’s circle theorem states that the radii of four mutually tangent circles \ (r_1, r_2, r_3, r_4\) satisfy. $$\begin {aligned} \left ( \frac {1} {r_1}+\frac {1} {r_2}+\frac {1} {r_3}+\frac {1} {r_4}\right) ^2 = 2\left ( \frac {1} {r_1^2}+\frac {1} {r_2^2}+\frac {1} {r_3^2}+\frac {1} {r_4^2}\right) \,. \end {aligned}$$
