Hyperbolic
Excenter
The excenter of a triangle is the intersection point
of the three external angle bisectors.
We can have three hyperbolic excenters
for a fixed triangle.
If
we think the external angle bisector as a line instead of a ray it can
exist till three intersection points. Then for each triangle can exist
till three excenters and three excircles.
Each excircle will be tangent either to one side of the triangle or to
its prolongation.
To construct the excenters, fixed
the triangle, we plot the hyperbolic line
that joins each pair of
vertices. Then we plot the hyperbolic angle
bisector of the
complementary
angle to obtain the external angle bisectors. As the sketch
that allows us to construct the hyperbolic bisector gives us a ray it is necessary to construct two
angle bisectors for each angle.
Once we have the 6 bisectors we drag the vertices so that the three
intersections exist. In this way we made sure that whenever one
intersection exist it will be plotted.
Triangles
Hyperbolic
geometry
