proof of hyperbolic circle theorem 1

hyperbolic circle theorem 1
Suppose that A and B lie on a hyperbolic circle K.
Then the segment AB divides K into two arcs.
Let C and D be points of K distinct from A and B.
Suppose that AB is not a diameter of K.
(1) C,D lie on the same arc if and only if E(A,B,C)=E(A,B,D).
(2) C,D lie on opposite arcs if and only if E(A,B,C)=-E(A,B,D).
proof
We may assume that the figure has been transformed so that
K has hyperbolic (and hence euclidean) centre O.

From the euclidean results (5) and (6), C,D lie on the same
arc if and only if e(ACB) = e(ADB). Since the euclidean angles
are in the range 0 to π, this is equivalent to cos(e(ACB))=cos(e(ADB)).
By the hybrid angle theorem and the definition of E, this is
equivalent to E(A,B,C) = E(A,B,D). This proves (1).

The proof of (2) is similar, but uses the fact that, in the range involved,
e(ACB)+e(ADB)=π if and only if cos(e(ACB))=-cos(e(ADB)).

hyperbolic geometry