Proof
Otherwise, we produce a point R and a circle C' with centre R such that
If P = O, then we may take as the h-inversion (the restriction of) reflection in the real axis.
We draw:
centre R, radius |RQ|. As the radii RQ and OR are perpendicular, the circles C and C' are orthogonal.
Observe that <OPQ and <OQR Then
Note that as O and P are inverse, each is mapped to the other by the h-inversion.
Finally, observe that the above argument
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If P is outside C, there is no h-line. |