Proof
If P = O, then we may take as the hinversion (the restriction of) reflection in the real axis.
Otherwise, we produce a point R and a circle C' with centre R such that
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 hinversion.
Finally, observe that the above argument

If P is outside C, there is no hline. 