

The Angle Bisectors Theorem for Hyperbolic Triangles The internal angle bisectors of a hyperbolic triangle are concurrent.
Proof
By the Hyperbolic Sine Rule applied to the htriangles AQC and AQB,
SImilarly, using the bisectors BR and CP,
Multiplying the ratios,
The angle bisectors all lie within the htriangle, so any two must meet.


Proof of the incentre property Suppose that X lies on the bisector of <BAC. Then hinversion in the hline AX maps points on AB to points on AC.
Let the hyperbolic perpendiculars from X meet AB in D and AC in F.
Now suppose that X also lies on the bisector of <BCA. Then we also It follows that D, E and F lie on an hcircle C with hcentre X.
Finally, as XD is perpendicular to AB, C touches AB at D, and
