This is the

valid in hyperbolic as well as euclidean geometry.

If we look at the lines through the vertices rather than just the segments, we can draw three further circles, each

touching all three lines. These are the

These are illustruated in the CabriJava applets below. You can move the vertices A, B, C to verify the results.

Apollonius asked what happens if we replace *some* of the lines with circles. In the language of inversive geometry,

we might ask the following questions:

Given three distinct i-lines, can we find an i-line touching all three?

If so, how many such i-lines are there?

The problem is often extended by allowing "point-circles" - i.e. a point may be regarded as a "circle" of zreo radius.

These do not fit neatly into the inversive description. We shall deal with this extension separately.

We shall see that:

There are many configurations where there are no such i-lines.

There is one type where there are an infinite number of i-lines.

Otherwise, the number may be 2, 4, 6 or 8.

The euclidean result mentioned about says the if we have three *non-parallel* extended lines, then there are four

*circles* which touch all three. Observe that two extended lines meet at ∞, so 'touch' if and only if this is the *only*

intersection, *i.e.* the corresponding lines are parallel. Since we have non-parallel lines, there is no extended line

touching all three. Thus, there are exactly four i-lines in this case.

**The euclidean result**

The incircle | |

One of the three excircles |