The Hyperbolic Tangents Theorem
If P lies on the h-circle K = K(Q,r), then
(1) there is a unique h-tangent to K at P, and
(2) the h-tangent is perpendicular to the h-line PQ.
Proof
By the Hyperbolic Perpendiculars Theorem, there is an h-line L
through P perpendicular to the h-line M = QP.
Suppose that L meets K again at R ≠ P.
Since M is perpendicular to L and passes through Q,
the h-centre of K, L and K are symmetric in M.
Thus, S, the inverse of R with respect to M, is on L and K,
so they meet three times, in R, P and S.
This is impossible (Basic Fact (2)), so L meets K once.
Thus L is an h-tangent at P.
By the Origin Lemma, there is an h-inversion t mapping P to O.
By the Hyperbolic Circles Theorem, t(K) is a circle through O.
Any h-line through O is a euclidean segment, so there is
exactly one,
which meets t(K) once (the euclidean tangent).
This must be t(L).
Hence the h-tangent is unique.
Part (2) follows from our construction of L.
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