Inversion in a circle
Our version of Apollonius's Theorem led us to make the
Definition We denote the inverse by i_{C}(P), or, if C is obvious, by P'.
Inversion is built into Cabri. 

Basic properties
There are, however, two problems: 
These occur since we cannot have OO.OP = r^{2}. 
To motivate the way we resolve this, you can use the CabriJava pane to investigate what happens as P approaches O. You should conclude that P' moves rapidly out along the ray OP. In an intuitive sense, P' approaches infinity. In fact, when P reaches O, The ray (and hence P') are undefined. 
See The extended plane 
Main Inversive Page 