Definition An haltitude of a hyperbolic triangle is an hsegment through a vertex perpendicular to the opposite side.
The Altitudes Theorem for Hyperbolic Triangles


Proof Let ABC be an htriangle, and let the haltitudes be AQ, BR and CP.
Since the labelling of the vertices is immaterial, we may as
If <ABC is a right angle, then AB and CB are haltitudes,
In the CabriJava figure on the right, we can move B to see
This is an easy consequence of the very important
Thus, we see that, either all the hratios are positive, or just
Consider the case where the angles are acute.
If the angle at B is obtuse, then we need to replace
Thus, by the Converse of Ceva's Theorem,


