Inversive Distance

The inversive distance is the Natural Logarithm of the ratio of two concentric circles into which the given circles can be inverted. Let $c$ be the distance between the centers of two nonintersecting Circles of Radii $a$ and $b<a$. Then the inversive distance is

$$\delta=\cosh^{-1}\left\vert{a^2+b^2-c^2\over 2ab}\right\vert$$

(Coxeter and Greitzer 1967).

The inversive distance between the Soddy Circles is given by

$$\delta=2\cosh^{-1}2,$$

and the Circumcircle and Incircle of a Triangle with Circumradius $R$ and Inradius $r$ are at inversive distance

$$\delta=2\sinh^{-1}\left({{1\over 2}\sqrt{r\over R}\,}\right)$$

(Coxeter and Greitzer 1967, pp. 130-131).

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 123-124 and 127-131, 1967.