The inverses of (extended) lines and circles
To make the statement of the following theorem as simple as possible,
we use the notions of the extended lines.
The Inversion Theorem
Let C be a circle with centre O.
 
Proof of the Inversion Theorem  There is a neater (algebraic) proof
on the Algebraic Inversion Theorem page 
CabriJava illustrations of cases 2 and 4


This shows the case where we are inverting an extended line L in the Circle C. If you move Q (on L), you will see that the image always lies on the circle L'.

This shows the case where we are invering a circle D in the circle C. If you move P on D, you will see that P' always lies on the circle D'.

The importance of this result is that it shows that, if J is either a circle or an extended line,
then the inverse will be a circle or extended line.
Thus, the Theorem says that inversion maps ilines to ilines!
These ilines play much the same role in inversive geometry thet lines do in euclidean geometry.
We now look at the the effect of inversion on angles between ilines.
Main Inversive Page 