We say that an i-line

The problem is to decide how many i-lines touch

As in the problem with three i-lines, we may invert to get a configuration where all the i-lines are circles.

We then have four basic cases, depending on the number of i-lines involved - 0, 1, 2 or 3. Of course, the

case of three i-lines has already been considered. The remaining cases each involve at least one point.

If **N** is a point, then we may apply a further inversion in a circle with centre **N**, so that we can assume

that **N** is the point ∞. Then any i-line "touching" this is an extended line. Since we assume that **N** does

not lie on the other "circles", **L**, **M** are still points or circles.

- Three points.

There is precisely one extended line through the*points***L**and**M**. - Two points and one circle.

Suppose that**L**is a circle, and**M**a point.

We require extended lines through**M**touching**L**.

If**M**lies inside**L**, then there are no such extended lines.

If**M**lies outside**L**, then there are two extended lines - the tangents to**L**through**M**.

Thus, we have either no common tangents to**L**,**M**,**N**or exactly two. - One point and two circles.

Suppose that**L**and**M**are circles.

We require (extended) lines tangent to both.

- One of
**L**,**M**lies strictly inside the other.

There can be no common tangents in such a case. -
**L**,**M**touch internally.

The only common tangent is that at the point of contact. -
**L**,**M**touch externally.

Then there are three common tangents (including that at the point of contact). -
**L**and**M**meet at two points.

Then there are two common tangents. - Each of
**L**,**M**lies ouside the other.

Then there are four common tangents.

- One of