Hyperbolic
circumcenter
Given a triangle, the circumcenter is defined as the point where the
three perpendicular bisectors intersect.
To construct the hyperbolic circumcenter of an arbitrary triangle it is
only
necessary to consider three points, to
plot the triangle with the hyperbolic triangle
tool and each one of the
perpendicular bisector with the hyperbolic perpendicular
bisector tool. The intersection, if it exists, is the circumcenter.
We can find that the three perpendicular bisectors do not intersect. If
that happens we drag one of the points of the
triangle until they intersect.
In this new situation we will be able to mark the intersection point.
Then, the intersection will appear if and only if it exists.
We observe that the point we obtain from the intersection of
the perpendicular bisector fulfills that it is at the same distance
from each one of the three points of the triangle. This is certain
because, as in the Euclidean Geometry, we have that the
perpendicular bisector is the geometric place of the points that
equidist from two given points. So, if we have a point that belongs to
three perpendicular bisectors it will be equidistant of each vertice.
The circumcenter is, therefore, the
center of the circumference that
passes through the three vertices. So, to study when exists the
intersection is the equivalent to study when there is a circumference
that goes through the three points. In Workshop
of Hyperbolic Geometry.pdf (in Catalan) we have classified the
different
situations which can be found. In the hyperbolic case it can happen
that
given three points, not lying in the same line, exists a circumference containing the
three points (existence of the circumcenter), a horocycle (existence of the circumcenter in the boundary line) or an
equidistant (no existence of the circumcenter).
From this tool it is interesting to look for a relation between the
three points that allows us to ensure if the circumcenter will exist.
You can find a characterization depending on the length
of the sides and on the major angle in Existence
of the circumcenter.
Triangles
Hyperbolic
geometry