Proof of the Perpendicular Theorem

The Perpendicular Theorem If H is an h-line, then,
for any point P, there is a unique h-line through P perpndicular to H.

Proof
As usual, we take the h-inversion t mapping P to O.
We show that there is a unique h-line K through O (i.e a diameter of C)
perpendicular to t(H).
Since inversions preserve angles, t^-1(K) is the required h-line.

First suppose that P lies on H.
Then as t(P) = O, t(H) is a diameter.
Since there is exactly one diameter K perpendicular to t(H),
the result holds in this case.

Now suppose that P does not lie on H.
Then t(H) is an arc of a circle C'.
Observe that a (euclidean) line L is perpendicular to C'
if and only if L passes through Q, the centre of C'.
Since there is exactly one diameter K of C through Q,
we get the result here also.

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