The Mirror Property
Suppose that L and M are distinct i-lines, then
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Proof Since L, M are distinct, M must contain points not on L. Since inversion in L maps points on one side of L to points on the other side, M must contain points on either side of L. It follows that, if iL(M) = M, then M must cut L twice in E+. Suppose that they meet in A and B.
As in the proof of the Uniqueness Theorem, we invert in a circle C with centre A.
Now consider the general case:
By the third part of the Algebraic Inversion Theorem,
iC° iL = iL'° iC, |
Algebraic Inversion Theorem |