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Let C1, C2, and C3 be mutually tangent circles with radii r1, r2, and r3. Let us assume that the radius of a fourth circle tangent to the other three, the red circle in Figure 1, is r4.1 Descartes’s circle theorem asserts the following:
We present a short proof of the Descartes circle theorem on the “curvature-centers” of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres...
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.
An article [7] published in this journal in 2019 gives a short history of the theorem and provides an original and straightforward proof based on Heron’s formula. Here, we provide an even more straightforward proof based on a generalization of Heron’s formula, the Cayley–Menger determinant.
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 ...
21 Οκτ 2019 · A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found.
A formula for the radii and positions of four circles in the plane for an arbitrary linearly independent circle configuration is found. Among special cases is the recent extended Descartes Theorem on the Descartes configuration and an analytic solution to the Apollonian problem. The general theorem for n-spheres is also considered.
