Orthogonal ilines
The Uniqueness Theorem
If A and B are distinct points on an iline L, then there is a unique iline M

Since the proof is a good example
of the use of inversion, we give it in full. 
Proof
Since A and B are distinct, one of them (A say) is not Ñ. Note the structure of the proof.


In a similar way we can prove a result fundamental to the creation of hyperbolic geometry.
The Mirror Theorem
Suppose that L and M are ilines, then 

Main inversive page 