The Algebraic Inversion Theorem
We use complex numbers as "coordinates" in E. The point A(a,b) is described by the complex number a =a + ib.
Then the locus A_{k}(A,B) = {P : PA = k.PB}
has equation
z a = k.z  b, For typographical reasons, we will use z* to denote the complex conjugate of z.
Immediately we see that inversion in the iline given by the real axis is 
We shall simply say that A is the point a 
Inversion in the iline z = r is also easy to describe,


We are now in a position to give a neat proof that inversion maps ilines The Algebraic Inversion Theorem
Suppose that L and C are ilines, and that i_{C} denotes inversion with respect to C.
Proof of The Algebraic Inversion Theorem


We can now prove a result fundamental to the creation of hyperbolic geometry.
Observe that, if L and M are lines, then i_{L} is just reflection in L.
In fact, this is true for any ilines. As you may guess, the proof amounts The Mirror Property
Suppose that L and M are distinct ilines, then


In fact, when , L and M are orthogonal, more is true.
If L and M are orthogonal ilines, then i_{L}(Mº) = Mº,
In the CabriJava pane on the right, S is a point on the arc of L lying inside M. You can experiment by dragging S or A.


We can think of this as saying that the two regions of the disc Mº are "reflections" of one another in a circular "mirror" L. 
This is not optically sound, but it is a good analogy. 
Main inversive page 