Touching Circles

In the introductory page, we mentioned a problem
about circles touching three mutually tangent circles.
We are now in a position to give a proof.


If three circles R, S and T touch at A, P and Q as shown,
then there are exactly two circles which touch all of them.

Of course, the proof consists of inversing in a circle
to make the picture simpler.

Let C be a circle with centre A.
Inverting in C sends A to Ñ, so R and S
map to extended lines R' and S'.
As A is not on T, T must map to a circle T'
touching R' at P and S' at Q'.

Any circle touching R' and S' must be have its center
on the bisector of P'Q', and radius ½P'Q'.
so there are exactly two touching T'.

Inverting again in C, we get the circles touching R, S and T.

Main inversive page