Forum Geometricorum, 6 (2006) 149--166.

Abstract: In Euclidean geometry the vertices P of those angles APB of size alpha that pass through the endpoints A, B of a given segment trace the arc of a circle. In hyperbolic geometry on the other hand a set of equivalently defined points P determines a different kind of curve. In this paper the most basic property of the curve, its convexity, is established. No straight-forward proof could be found. The argument rests on a comparison of the rigid motions that map one of the angles APB into other ones.

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