Hyperbolic segments and triangles

In euclidean geometry, we have the idea of a segment AB, consisting
of A, B and all points of the line AB which lie between A and B.

Suppose that P and Q lie on an h-line H. The points define an arc of H.
If tεH(2), then, by Theorem 1, t(P) and t(Q) lie on the h-line t(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 = L_nD, and let t* be the element of I(2)
corresponding to t. From the i-Segment Theorem, t* must map
the arc PQ to one of the arcs of t*(L) cut off by t*(P) and t*(Q),
Since t* maps D to D, it must map PQ to the arc of t*(L) in D,
i.e. to the arc of t(H) defined by t(P) and t(Q). Thus, we have

Theorem 2
If R is between P and Q, and tεH(2), then t(R) lies between t(P) and t(Q).

Definition
If P and Q are distinct points of D, then the h-segment PQ consists of
P, Q and all points which lie between P and Q.

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
is just a euclidean segment.

Suppose that P, Q and R are points which do not lie on an h-line.
The h-segments, PQ, QR and RP constitute the h-triangle PQR.

The figure on the left shows an h-triangle. You can move the vertices
to change the shape and size.

As in euclidean geometry, we regard the angles of a triangle as unsigned
quantities in the range (0,π). Of course, the angle between arcs is
defined in terms of tangents.

We can now prove a result quite different from euclidean geometry.

The Hyperbolic Triangle Theorem
The sum of angles of a hyperbolic triangle is less than π.

Proof of the hyperbolic triangle theorem

Suppose we produce the h-segment PQ beyond Q, then we have an
exterior angle at Q. If the interior angle is θ then the exterior angle
has magnitude π-θ.

It is an easy exercise to deduce the

Corollary
The exterior angle of an h-triangle is greater than
the sum of the interior opposite angles.