In euclidean geometry, we have the idea of a segment AB, consisting
Suppose that P and Q lie on an h-line 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 i-line with H = LnD,
and let t* be the element of I(2)
Theorem 2 says that elements of H(2) map h-segments to h-segments.
Note that if the underlying h-line is a diameter, then the h-segment
Suppose that P, Q and R are points which do not lie on an h-line.
The figure on the left shows an h-triangle. 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
Suppose we produce the h-segment PQ beyond Q, then we have an
It is an easy exercise to deduce the