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
|One of the three excircles|