More about apollonian families
From Apollonius's Theorem, we know that the iline L belongs to the apollonian family A(A,B) if and only if A and B are inverse with respect to L. It follows that each iline belongs to infinitely many families.
Now suppose that M is a second iline. Then L and M
Since members of an apollonian family are disjoint, 
See Apollonius's Theorem

It turns out that such a pair {A,B} always exists. The Common Inverses Theorem.
If L and M are nonintersecting ilines, then
there exists a
The CabriJava pane shows the case of two circles.


As indicated in the preamble, we would like to restate this as
Theorem
But there is a problem.
Our definition of apollonian family, does not allow A or B to be Ñ.
Our version of Apollonius's Theorem offers a solution since it shows that the With this new definition, the Theorem becomes true.


The new definition also allows us to state without exclusions, the Theorem Inversion maps apollonian families to apollonian families.
This is quite simple. Thus i maps A(A,B) to A(i(A),i(B)).


We also have The Concentric CirclesTheorem
If L and M are nonintersecting ilines, then there is an inversion
We know, by the Common Inverses Theorem, that there are points A and B

This result is a major step in understanding Steiner's porism

Main inversive page 