Some results of Apollonius - properly intersecting circles

Suppose that L and M meet at P and Q.
If P or Q lies on N, then we can invert in a circle with the common point as centre.
This gives a configuration with three extended lines. We have dealt with such cases.

We may now suppose that Q is not on N. By inverting in the circle through P with centre Q,
we can assume that L and M are extended lines meeting at P, and that N is a circle.
The circle N cuts L and M twice each.

Observe that no extended line can touch the non-parallel lines L and M.
Also, a circle touching these lines has its centre on a bisector of one of the angles made by
the lines L and M. The lines L and M divide the plane into four sectors, and each of the
tangent circles lies in one of the sectors.

The point P cuts L into two half-lines. If N cuts both, then P lies inside N, so that N also cuts
both parts of M. Otherwise, N cuts one of the half-lines of each of L and M.

We can use a CabriJava applet to investigate these two cases:

1. P lies inside N.
   Here, L and M divide N into four arcs, one in each sector.
   Considering the tangent circles in each sector, we get two common tangents in each
   sector, making eight in all.
2. P lies outside N.
   Now, L and M divide N into four arcs, two in one sector, one in each adjacent sector.
   Again each arc is touched by two of the tangent circles. (Note that, if a circle touches
   one arc of N, then it meets N nowhere else.) Again, we have eight common tangents.

touching circles

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