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24 Ιαν 2022 · 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 0 0 1 1 b 1 = b 2 = 0, b 3 = b 4 = 1 (0 + 0 + 1 + 1)2 = 22 = 4 2(02 + 02 + 12 + 12) = 2(2) = 4 Edna Jones The Descartes circle theorem
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.
DESCARTES' CIRCLE THEOREM. If there exist three circles (C1, C2, C3, in black, below) that are mutually tangent externally and have radii r1, r2, r3, and a fourth circle (C4 in red, below - there are two possiblities) having radius r4 that is tangent to the first three, then the radii are related by ( 1 1 1 1. r.
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Descartes' Circle Formula is a relation held between four mutually tangent circles. Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius.
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 Descartes circle theorem is an ancient theorem about four mutually tangent circles in the Euclidean plane, asserting that their curvatures satisfy the quadratic relation, 2(κ2. + κ2. + κ2. + κ2 4) = (κ1 + κ2 + κ3 + κ4)2 .
