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.