# Hyperbolic segments and triangles

 hyperbolic segments 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 = LnD, 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. CabriJava Window hyperbolic triangles 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. 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. CabriJava Window