Proof of the Hyperbolic Mid-point Theorem

The Hyperbolic Mid-point Theorem
If P and Q are distinct points of D, then there is a unique point R
on the h-segment PQ such that D(P,R) = D(Q,R).

Proof of the Theorem
Since hyperbolic transformations map h-segments to h-segments,
and preserve D, an easy application of the Origin Lemma shows that
we may consider just the case of an h-segment OA.

In this case, the Origin Lemma shows that there is an h-inversion h which
maps A to O and O to A. Suppose that this is h-inversion in the h-line H.
Since O and A must lie on opposite sides of H, the h-line H
must cut the h-segment OA. Suppose H and OA meet at B.
Since B is on H, h fixes B, and, by our choice of h, h(O) = A.
Then, by the Corollary to the Hyperbolic Circles Theorem,
D(O,B) = D(h(O),h(B)) = D(A,B).

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