More about apollonian families

From Apollonius's Theorem, we know that the i-line 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 i-line belongs to infinitely many families. Now suppose that M is a second i-line. Then L and M will belong to the same apollonian family A(A,B) if and only if A and B are inverse with respect to L and to M. Since members of an apollonian family are disjoint, this can only happen if L and M are non-intersecting.	See Apollonius's Theorem
It turns out that such a pair {A,B} always exists. In fact, the pair is uniquely determined by the i-lines. The Common Inverses Theorem. If L and M are non-intersecting i-lines, then there exists a unique pair of points {A,B} inverse with respect to both L and M. The CabriJava pane shows the case of two circles. If you drag the circles or there centres the figure will change. You can check that A and B disappear when L and M meet Proof
As indicated in the preamble, we would like to restate this as Theorem A pair of non-intersecting i-lines determine a unique apollonian family. But there is a problem. The proof in the case of two circles involves an inversion. When we invert back to recover the original picture, one of the points may map to Ñ. Our definition of apollonian family, does not allow A or B to be Ñ. This is because the lengths ÑA and ÑB are undefined. Our version of Apollonius's Theorem offers a solution since it shows that the apollonian family A(A,B) consists of i-lines for which A and B are inverse points. Now there is no need to exclude Ñ. The points A and Ñ are inverse with respect to an i-line L means that iL(A) =Ñ. Thus the family A(A,Ñ) is the family of (concentric) circles with centre A. 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. Suppose we let i denote inversion with respect to an i-line C. From the Algebraic Inversion Theorem, A and B are inverse with respect to L if and only if i(A) and i(B) are inverse with respect to i(L). Thus i maps A(A,B) to A(i(A),i(B)).
We also have The Concentric CirclesTheorem If L and M are non-intersecting i-lines, then there is an inversion mapping them to concentric circles. We know, by the Common Inverses Theorem, that there are points A and B inverse to L and M. One of these (A, say) is not Ñ. Inversion in a circle C, centre A sends A to Ñ, so sends L and M to circles with centre iC(B).	This result is a major step in understanding Steiner's porism

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