The Mirror Property
Suppose that L and M are distinct ilines, then


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 i_{L}(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,
i_{C}° i_{L} = i_{L'}° i_{C}, 
Algebraic Inversion Theorem 