Figure 8: Construction of the hyperbolic circle c2 with hyperbolic center A ≠ O and passing through B for the case in which points A, B and O are not collinear. All hyperbolic lines passing through the center A are orthogonal to the circle. Since line OA is one such line, it is orthogonal to c2 and the center of c2 must lie on this line (if a line cuts a circle orthogonally, the line must be a diameter of the circle). In addition, c2 is orthogonal to the hyperbolic line pAB (circular arc) passing through points A and B . By Theorem 1, the Euclidean center O2 of c2 must lie on the tangent tB to the hyperbolic line pAB at point B . Thus the Euclidean center O2 of c2 is located at the intersection of line OA and tangent tB. The hyperbolic circle c2 can then be constructed using a Euclidean circle centered at O2 and passing through B.
Begin by hiding c1 from the previous case and then implement the construction using the following GeoGebra code. Although the most obvious way to construct the tangent is to use the Tangent[ <Point>, <Conic> ] command, we forego this approach since the command fails for some locations of B on γ (returns undefined, most likely since the numerical algorithms occasionally fail to find the tangent at the endpoint of an arc). The approach taken here is to use the fact that the tangent to a circle is perpendicular to its radius.