Given two points in the upper half-plane with the usual hyperbolic metric, the geodesic between them is found by drawing a circle through them that crosses the real axis at right angles. However, if I give you coordinates for the points, how do I construct such a circle explicitly? I guess this becomes a question in classical plane geometry, but I don't see how to do it.
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The center of the circle lies on the real axis, and also on the perpendicular bisector of the two points, so...