Never an ugly sight in geometry

God made all things beautiful......

Descartes' Theorem

If four mutually tangent circles have curvature k_i (for i = 1,…,4), Descartes' theorem says:

(1)

When trying to find the radius of a fourth circle tangent to three given kissing circles, the equation is best rewritten as:

(2)

The ± sign reflects the fact that there are in general two solutions. Other criteria may favor one solution over the other in any given problem.

The curvature mentioned earlier is the 'bend' in the poem below, and it refers to the reciprocal of the radius. (1/r)

The Kiss Precise

For pairs of lips to kiss maybe
Involves no trigonometry.
This not so when four circles kiss
Each one the other three.
To bring this off the four must be
As three in one or one in three.
If one in three, beyond a doubt
Each gets three kisses from without.
If three in one, then is that one
Thrice kissed internally.

Four circles to the kissing come.
The smaller are the benter.
The bend is just the inverse of
The distance from the center.
Though their intrigue left Euclid dumb
There's now no need for rule of thumb.
Since zero bend's a dead straight line
And concave bends have minus sign,
The sum of the squares of all four bends
Is half the square of their sum.

To spy out spherical affairs
An oscular surveyor
Might find the task laborious,
The sphere is much the gayer,
And now besides the pair of pairs
A fifth sphere in the kissing shares.
Yet, signs and zero as before,
For each to kiss the other four
The square of the sum of all five bends
Is thrice the sum of their squares.

- Frederick Soddy