Some results of Apollonius - two disjoint circles
Suppose that L and M are disjoint circles, and that N is a third circle which does not touch L or M.
By the concentric circles theorem, we can apply an inversion so that L and M become concentric.
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
- 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.
- N lies outside M, and M and outside N.
Inversion in N reduces it to a nested case - again there are no tangents.
- 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.
- N lies inside M, and cuts L twice.
Here, each of the families contains just two tangents,
so there are four in all.
- N lies outside L, and cuts M twice.
Inversion in L reduces this to the previous case,
so again we have four tangents.
- 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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