The Algebraic Inversion Theorem
Suppose that L and C are i-lines, and that iC denotes inversion with respect to C.
We observe that the definition of inversion depends only on length,
so does not depend on the choice of axes.
Thus, if C is an extended line, then we may assume that it is the real axis, so iC(z)=z*.
If C is a circle, we may assume it is |z|=r, so iC(z)=r2/z*.
Suppose that A and B are inverse with respect to L,
If C is the real axis, then iC(L) has equation
|z*-a| = k|z*-b|
If C is the circle |z|=r, then iC(L) has equation
|r2/z*-a| = k|r2/z*-b|.
Suppose that B is the inverse of A with respect to L, i.e. that B = iL(A).
|The Algebraic Inversion Theorem|