In this section, we establish basic results about hyperbolic lines and angles. This includes results illustrated in the experimental pages.
Taking the kleinian view, we base our development of the theory
Suppose that H is an hline. We refer to the restriction of i_{L} to D as hinversion with respect to H, denoted by i_{H}.
The hinversions play a role in hyperbolic geometry very similar to the role


Theorem 1 If t is a hyperbolic transformation, then


In our Poincare disk model, O appears to have a special status, since
the hlines through O are euclidean segments rather than arcs of circles. In fact, it is typical, as we shall see.
Origin Lemma
The CabriJava pane on the right illustrates the Lemma. More generally, we have the
Interchange Lemma Proof of the Interchange Lemma. Since the hinversion, and hence the hline, in the Interchange Lemma are unique, we make the
Definitions
The justification for the names hbisector and hmidpoint will become clear
These Lemmas can be used to prove fundamental results on The Incidence Theorem
Proof
Note that these are exact analogues of incidence results in
There are many analogues of euclidean results. We give some
The Angle Bisector Theorem
This is left as an exercise to the reader. Simply apply the Origin Lemma


The Perpendicular Theorem If H is an hline, then, for any point P, there is a unique hline through P perpendicular to H. Proof of the Perpendicular Theorem
In the figure, you can move the point P,


In euclidean geometry, we say that a figure P is symmetric about a line L if reflection in L maps P to itself.
Definition
Symmetry of hlines This is a consequence of the Mirror Property from inversive geometry.
We shall meet more interesting examples when we look at hyperbolic
