The Algebraic Inversion Theorem
Suppose that L and C are i-lines, and that iC denotes inversion with respect to C.
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Proof 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 |