Theorem of Descartes

In Geometry, the theorem of Descartes , discovered by Rene Descartes, a relation between four Cercle S tangent between them establishes. It can be used to build a fourth tangent circle to the three others.

History

The geometrical problems concerning of the tangent circles were discussed since of the millenia. In ancient Greece, 3 centuries before Jesus-Christ, Apollonius de Perga devoted a whole book to this subject. Unfortunately this book, Tangencies , disappeared.

Rene Descartes briefly speaks about the problem in 1643, in a letter addressed to the Elisabeth princess of Bohemia. He provided the same solution primarily as that given in the equation (1) below, this is why its name was given to the theorem.

Frederick Soddy has redécouvert the equation in 1936. The circles in this problem are sometimes known as a circles of Soddy , perhaps because Soddy chose to publish its version of the theorem in the form of poetry entitled The KIS specifies , which was printed in '' Nature '' the June 20th 1936. Soddy also extended the theorem to the Sphère S.

Definition of the curve

The theorem of Descartes is stated more simply by using the Courbure circle. The curve of a circle is thus defined k  =  ±1/r , where R is its ray. The larger the circle is, plus its curve is small, and vice versa.

The plus sign in K   =  ±1/ R is used for a circle which is tangent outside with the other circles, like the three black circles in the figure opposite. In the case of a tangent circle internally , like the large red circle in the figure, the minus sign is used.

The theorem of Descartes

If four tangent circles between them have as a curve K I (for I   =  1… 4), the theorem of Descartes states:

(k_1+k_2+k_3+k_4) ^2=2 \, (k_1^2+k_2^2+k_3^2+k_4^2).

Rewritten the equation gives us the radius of the fourth tangent circle with three circles given tangent:

k_4=k_1+k_2+k_3\pm2\sqrt{k_1k_2+k_2k_3+k_3k_1}.
The sign ± indicates that there exists two in general solutions. Other criteria can support a solution.

Particular case

If one of the three circles is replaced by a line, K 3 (for example) is null. Thus the equation (2), simplified, gives us:

k_4=k_1+k_2\pm2\sqrt{k_1k_2}.

The theorem of Descartes does not apply when more than one circle is replaced by a line. The theorem does not apply either when more than one circle is tangent internally, for example in the case of three tangent overlapping circles in a point.

Theorem complexes of Descartes

In order to define a circle completely, not only its radius (or its curve), but also its center must be known. The suitable equation is clearer if the coordinates ( X ,   ) is interpreted there like a Complex number Z   =  X   +  I there . The equation is then similar to the theorem of Descartes and is called the complex theorem of Descartes .

Are four circles of curve K I and of center Z I (for I   =  1… 4), the following equality draws from the equation (1):

(k_1z_1+k_2z_2+k_3z_3+k_4z_4) ^2=2 \, (k_1^2z_1^2+k_2^2z_2^2+k_3^2z_3^2+k_4^2z_4^2).

Once K 4 found via the equation (2), one can calculate Z 4 by rewriting the equation (4) in a form similar to the equation (2). Once again, there will be in general two solutions for Z 4, corresponding to the two solutions for K 4.

See too

  • Circle of Ford
  • Circle of Soddy
  • Circle of Apollonius

External bonds

  • Applet interactive showing four tangent circles.

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