Figure 9: Construction of the hyperbolic circle c3 with hyperbolic center A and passing through B for the case in which A ≠ O and A and B are collinear with O . Begin by determining the inverse of A with respect to γ and then construct circle ca passing through A and A ’ and centered at the midpoint of segment AA’. By Theorem 2, ca is orthogonal to γ and is therefore a hyperbolic line in the Poincaré disk. Define A” and B” as the inverses of A and B with respect to ca (be careful since in all previous constructions, the inverses have been with respect to γ). To finish the construction, we use the distance metric. By Theorem 5, d (A”, B”) = d(A, B). Furthermore since A is invariant after inversion with respect to ca , we obtain d(A, B”) = d(A, B ). Hence, the hyperbolic circle c3 can be constructed using a Euclidean circle centered at the midpoint of segment BB’’ and passing through the point B.
Begin by hiding all of the objects from the previous construction except points A and B and the defining circle γ and then move the A and B onto a diameter. Use the following GeoGebra code to implement the construction.