More theorems about hyperbolic triangles

The Converse of Ceva's Theorem allows us to prove
some familiar looking results in hyperbolic geometry

Definition
An h-median of a hyperbolic triangle is an h-segment
joining a vertex to the h-midpoint of the opposite side.

The Medians Theorem for Hyperbolic Triangles
The h-medians of a hyperbolic triangle are concurrent.

Proof of the medians theorem

The figure on the right is a CabriJava illustration
of the Medians Theorem. You can move the
vertices A, B and C (within the boundary circle).
The points P, Q and R are the h-midpoints of the
sides of the h-triangle ABC.


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
is the centre of the incircle - a circle touching all three sides. Indeed, the
same proof suffices.

Proof of the angle bisector theorem and the incentre property

In euclidean geometry, we then look at excircles - that is to say circles which
touch all three sides of a triangle, but lie outside the triangle. These are best
by looking at external bisectors of the angles. If we try this in the hyperbolic
plane, we may find that the h-lines do not intersect, and no excircle exists.

CabriJava illustration of excircles

In these theorems, the h-segments all lie within the h-triangle, so that any two
must meet. Then the Converse of Ceva's Theorem gives an unconditional result.

In situations where two of the h-segments lie outside the h-triangle, we can
have cases where the product of the h-ratios is 1, but the h-segments do not
meet. A simple example is that of (hyperbolic) altitudes.

Definition
An h-altitude of a hyperbolic triangle is an h-segment
through a vertex perpendicular to the opposite side.

The Altitudes Theorem for Hyperbolic Triangles
If any two h-altitudes of a hyperbolic triangle meet,
then the h-altitudes are concurrent.

Proof of the altitudes theorem

The figure on the right is a CabriJava illustration
of the Altitudes Theorem. You can move the
vertices A, B and C (within the boundary circle).
The points P, Q and R are the feet of the
h-altitudes of the h-triangle ABC.

By experimenting, you should be able to see that
(1) there are cases where no two meet, and that
(2) these occur only in certain cases where the
h-triangle has an obtuse angle.


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.

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