The Converse of Ceva's Theorem allows us to prove some familiar looking results in hyperbolic geometry
Definition
The Medians Theorem for Hyperbolic Triangles
The figure on the right is a CabriJava illustration


The Angle Bisectors Theorem for Hyperbolic Triangles The internal angle bisectors of a hyperbolic triangle are concurrent.
As in euclidean geometry, the point of concurrence of the angle bisectors Proof of the angle bisector theorem and the incentre property
In euclidean geometry, we then look at excircles  that is to say circles which CabriJava illustration of excircles
In these theorems, the hsegments all lie within the htriangle, so that any two
In situations where two of the hsegments lie outside the htriangle, we can



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 of the altitudes theorem
The figure on the right is a CabriJava illustration
By experimenting, you should be able to see that


Many other euclidean theorems have hyperbolic analogues. For example, Stewart's Theorem,
and van Obel's Theorem.
In experiments with circles, we looked at the existence of a hyperbolic circle
through three points  the circumcircle of a triangle. We can now study this in detail.