The Interior Exterior Theorem
Suppose that L is an iline. If L is an extended line, then it divides the rest of E^{+} into two regions, known as the sides of L. If L is a circle, then it also dvides the rest of E^{+} into two regions, known as the interior , the region containing the centre, and the exterior the region containing ∞. For the sake of uniformity, we shall also refer to these as sides of the circle.
We know that an inversive transformation t maps L to an iline,
The concept of region is essentially topological, but we wish to give a


The InteriorExterior Theorem. If L is an iline and tεI(2), then t maps the sides of L to the sides of t(L).
A nice way to show this would be to prove that if two points A and B lie


In euclidean geometry, the idea of a segment is best described using the idea of betweenness. However, we cannot say that, for three points on an circle, one lies between the other two in any meaningful way.
The InteriorExterior Theorem does allow us to define segments of ilines
Definition
The existence and uniqueness of such an M is guaranteed by
It might appear simpler to take any iline M through A and B, but then
If L is a circle, this produces the obvious result  the euclidean arcs of L. 

After the InteriorExterior Theorem, it is easy to see that inversive transformations map isegements to isegements. We have
The isegment Theorem 
Main inversive page 