Some results of Apollonius - two touching circles

Suppose that circles L and M touch at P.
If we invert in a circle with centre P, then L and M map to extended lines.
Since L and M touch at P, the lines are parallel.

If P also lies on N, then this also inverts to an extended line.
This leads to a case with three lines, considered earlier.

Thus we now consider cases where L and M are parallel extended lines, and N is a circle.
For definiteness, we imagine the lines as horizontal, with L below M.

An extended line "touches" L and M if and only if it is paralle to both. There are two such
lines touching N, but these may include L or M, which do not count as common tangents.
The circles touching both form a single family, with centres lying on the line equidistant
from each. They lie between L and M.

The CabriJava applet can be used to investigate the cases:

  1. N lies entirely below L, or entirely above M.
    Here none of the tangent circles meets N, the only
    two common tangents are the extended lines.
  2. N lies strictly between L and M.
    Now we have the two extended lines, and four circles,
    so we have six common tangents in this case.
  3. N lies between L and M, but touches one of them.
    Now only one of the extended lines is a common tangent (the other is L or M).
    Of the family of cicles, two touch N externally.
    If N touches L, say, at P, then the only member touching N internally is that
    touching L at P. Thus, we have four common tangents.
  4. N lies below L and touches it, or above M, and touches it.
    Again, only one of the extended lines is a common tangent (the other is L or M).
    If N touches L, say, at P, then the only member touching N is that
    touching L at P. Thus, we have two common tangents.
  5. N lies between L and M, and touches both.
    The extended lines are L and M, so neither counts as a common tangent.
    There are two members of the family of circles touching N externally.
    The only one touching "internally" is N itself (which does not count).
    Thus, there are two common tangents.
  6. N cuts one of the lines L, M twice, but does not meet the other.
    Here, we have the two extended lines as common tangents.
    There are two of the family of circles touching N externally, and none internally.
    Thus, there are four common tangents.
  7. N cuts one of the lines L, M twice, and touches the other.
    Here, only one of the extended lines is a common tangent (the other is L or M).
    There are two of the family of circles touching N externally, and one internally.
    Thus, there are four common tangents.
  8. N cuts each line twice.
    We have the two tangent extended lines, and four tangent circles.
    Thus we have six common tangents.

disjoint circles

intersecting circles

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