Some results of Apollonius - two disjoint circles

Note that if the centre of the inverting circle lies on N, then N maps to an extended line. We can get
round this by inverting in a circle inside the images of L and M whose centre does not lie on the image
of N. Then N inverts to a circle.

We now suppose that this has been done. Switching the labels L and M if necessary, we may assume :

Circle L lies inside circle M and circle N touches neither.

We immediately observe that no extended line can touch L and M since a tangent to M lies outside M.
Also, any circle which touches both must touch M internally. It may touch L internally or externally.
See the basic theorem. Note that the tangent circles lie btween L and M.

Thus, we have two families of circles touching L and M. These are illustrated by CabriJava.

Since the circle N does not touch either circle we have the cases

1. N lies outside L or inside L.
   Since N does not meet the region between L and M,
   there are no tangents in these cases.
   
2. N lies outside M, and M and outside N.
   Inversion in N reduces it to a nested case - again there are no tangents.
3. L lies outside N, but N lies inside M.
   It is easy to see that each of the two families contains four tangent circles.
   Thus, there are eight tangents in all.
4. N lies inside M, and cuts L twice.
   Here, each of the families contains just two tangents,
   so there are four in all.
5. N lies outside L, and cuts M twice.
   Inversion in L reduces this to the previous case,
   so again we have four tangents.
6. N cuts L and M twice each.
   Now the family of circles outside L contains four tangents to N
   The family of circles containing L contains no tangents, since
   such a circle contains the intersection of L and N, but not all of N.
   Thus, there are four tangents in this case.

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touching circles

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