Figure 7: Construct the hyperbolic line passing through points
A and B for the case in which A and B are
not collinear with O and both points lie on γ. Begin by
constructing tangents t3a and t3b to γ at points A
and B respectively. By Theorem 1, the tangents intersect at the
center O3 of a circle c3 which passes through A and
B and is orthogonal to γ. Construct c3 using O3 as
the center and either point A or B. Use A and
B as the endpoints of arc a3. Determine an additional
point M3 on arc a3 by constructing a ray from O3 to
the midpoint of segment AB and locating the intersection of this
ray with c3. Then use points A, M3, B to
construct arc a3.
Prepare for this construction by hiding all of the objects from the
previous constructions except for points A and B and the
defining circle γ. Then construct arc a3 using the GeoGebra code
provided below. Note that the most obvious method for constructing the
tangents would be to use the Tangent[ <Point>,
<Conic> ] command. However if this command is
used, the final hyperbolic line tool will fail when imported into
another GeoGebra file (most likely due to the fact that the Tangent
command was designed to construct multiple tangents from a point to a
conic). To work around this problem, unique tangents to γ at A
and B are constructed using the fact that the tangent to a circle
is perpendicular to the radius of the circle.