hyperbolic segments
In euclidean geometry, we have the idea of a segment AB, consisting
Suppose that P and Q lie on an hline H. The points define an arc of H. It is not clear the arc PQ will map to the arc defined by t(P) and t(Q).
Let L be the iline with H = L_{n}D,
and let t* be the element of I(2)
Theorem 2
Definition Theorem 2 says that elements of H(2) map hsegments to hsegments.
Note that if the underlying hline is a diameter, then the hsegment


hyperbolic triangles
Suppose that P, Q and R are points which do not lie on an hline.
The figure on the left shows an htriangle. You can move the vertices
As in euclidean geometry, we regard the angles of a triangle as unsigned We can now prove a result quite different from euclidean geometry.
The Hyperbolic Triangle Theorem Proof of the hyperbolic triangle theorem
Suppose we produce the hsegment PQ beyond Q, then we have an It is an easy exercise to deduce the
Corollary

