Descartes' theorem

Remarks

The numbers $k_i$ are the signed curvatures of the circles. The statement extends to the case where one circle is a straight line, with curvature $0$. Solving the quadratic equation for $k_4$ gives $$k_4 = k_1 + k_2 + k_3 \pm 2\sqrt{k_1 k_2 + k_2 k_3 + k_3 k_1},$$ and the two signs correspond to the two circles that touch $\Gamma_1, \Gamma_2, \Gamma_3$. Example: an outer circle of radius $1$ ($k_1 = -1$) holds two circles of radius $1/2$ ($k_2 = k_3 = 2$). The formula gives $k_4 = 3$, and indeed $(-1 + 2 + 2 + 3)^2 = 36 = 2(1 + 4 + 4 + 9)$. One proof applies an inversion that sends two of the circles to parallel lines and computes directly.

Going deeper

For a student

The theorem is the basic step in the construction of Apollonian circle packings. Starting from four mutually tangent circles, the second root of the quadratic replaces one circle by a new one, and the process repeats. If the first four curvatures are integers, all curvatures in the packing are integers, which makes these packings an object of number theory. It is a special case of the problem of Apollonius where the given circles touch each other.