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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 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
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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.
A Descartes configuration of circles consists of four mutually tangent circles in the plane. Descartes was con-cerned with configurations such as that at A in the above display, with bend, or curvature, defined as the reciprocal of radius. For example, the circles at A might have radii 1/4, 1/12, 1/13 and 1/61.
Workout. Question 1: Prove that the angle in a semi-circle is always 90°. Question 2: Prove that the angle at the centre is twice the angle at the circumference. Question 3: Prove the angles in the same segment are equal. Question 4: Prove the opposite angles in a cyclic quadrilateral add to 180°.
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 .
Revision for this topic. Prove that the angle in a semi-circle is always 90°. Prove that the angle at the centre is twice the angle at the circumference. Prove the angles in the same segment are equal. Prove the opposite angles in a cyclic quadrilateral add to 180°.
