# Some results of Apollonius - other cases with three extended lines

Suppose that L and M are parallel lines.
Then it is easy to see that any circle touching both must lie between the lines.
Also, an extended line touching both must parallel to both, but not identical to either.

If M is a third line, then there are two possibilities:

• M is parallel to both.
Relabelling the lines if necessary, we may assume that N is between L and M.
Then any circle touching L and M will cut N twice.
Any extended line touching L and M must be parallel to both, and hence also to N.
Here we have an infinite family (the i-lines parallel to L, M and N, but excluding these three)
• M cuts both, at A, B, say.
The CabriJave applet below shows this situation.
There are clearly two circles which touch all three lines.
There are no extended lines since any parallel to L and M will cut N at a point other than ∞.

Thus, with three extended lines, we always have touching i-lines.
There may be two, four or infinitely many.

You can vary the line AB by moving A or B.
You can vary the circle by moving C.
If you drag C from one side of AB to the other, the required circles are clear.

CabriJava illustration