I supose you're working with the Poincare's half (upper) plane model of a dimension 2 hyperbolic geometry,and thus you'd need a
half circle through with center on the real axis. There are infinite such circles, of course, but here's a
geometric easy way to choose a "nice" one: find the equation of the perpendicular bisector of the segment determined by
, and find its intersection with the real axis, which will be our half circle's center...