Then it is easy to see that any

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
**