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:

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

general results

main apollonius page