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We study the curvature and isometries of the quasihyperbolic metric on plane domains. We prove that, except for the trivial case of a half-plane, the isometries are exactly the similarity mappings. We need to assume that the boundary of the domain is C 3 smooth. MSC2000: 30F45.doi:10.2140/pjm.2007.230.315 fatcat:esbb2i52evguzapnkgyz7pr63m