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. The theorem was first stated in a 1643 letter from René Descartes to Princess Elizabeth of the Palatinate, presumably in an attempt to impress her.
Suppose circles for are mutually tangent, i.e., any two of the four circles touch at precisely one point. Let denote the radius of .
The curvature of is defined to be where the sign of is chosen depending on whether or not is internally or externally tangent to the other circles. For example, in the above picture, the large green circle has negative curvature, since the other three circles are internally tangent to it. On the other hand, the small red circle has positive curvature, since the other three circles are externally tangent to it.
Descartes' circle theorem states that the following quadratic equation holds:
In particular, if , , and are known, one can solve for as
These are the two explicit possibilities for . One of these solutions is positive, and the other is either positive or negative; if the second solution is negative, it must represent a circle that is internally tangent to the other three (like the large green circle in the diagram above).
A straight line can be thought of as a circle with infinite radius, or equivalently, zero curvature. Thus, it is possible to set one of the curvatures in Descartes' equation equal to zero and obtain the relation describing a situation like the below picture:
Setting , this relation is the simpler equation . For example, using the numbers from the diagram above, one obtains , as depicted.
Although Descartes' circle theorem is easily stated, it is not easy to prove. Below is a proof via trigonometry:
We start with the trigonometric identity that states if , then Proof of identity: Rewriting the above identity, we have We will show that We start with Now, Then
This implies i.e. Hence, we finally get our identity.
Now, triangle in the diagram to the right, where and are the centers of the three black circles, has side lengths Let be the center of the small red circle with radius externally tangential to all of the three black circles, then Also, let Then applying the cosine rule in triangles and gives Replacing radii with their respective curvatures, we get Similarly, we have Placing the values of on our trignometric identity, we get Simplifying, we get Dividing both sides by gives Putting the values of gives Simplifying, we get Further simplifying, we get Finally, we have Hence, the theorem is proved.
Descartes' circle theorem can also be generalized to non-2D space, in a theorem sometimes referred to as the Soddy-Gosset theorem. In Euclidean space of dimensions, the maximum number of mutually tangent spheres is . In 2-dimensional space, that's 4 mutually tangent circles; in 3-dimensional space, 5 mutually tangent spheres can always be found. The relationship of their curvatures is