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Basic Properties of Hyperbolic Ratio

(1) h(A,B,X) = -h(B,A,X),
(2) if X is between A and B, then h(A,B,X) ε (0,1),
(3) if X is on AB, beyond B, then h(A,B,X) ε (-∞,-1),
(4) if X is on AB, beyond A, then h(A,B,X) ε (-1,0).

Note that the value of h(A,B,X) determines the position of X relative to A and B.

Proof
(1) is an obvious consequence of the definition of h.
(2),(3) and (4) all follow from property D3, and the fact that
the function sinh is increasing.

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