 Every hyperbolic-plane isometry is a product of at most three reflections

A reflection in the perpendicular bisector of a point and its image corrects that point. A second bisector reflection through the corrected point can correct another point without moving the first. An isometry fixing two distinct points is either the identity or reflection in their joining geodesic: points with prescribed distances from both centres have at most two possibilities, exchanged by that reflection. Fixing a third noncollinear point then fixes every point. This constructs a factorization with at most three hyperbolic reflections.