Mappings preserving the area equality of hyperbolic triangles are motions

Abstract

It is shown that a mapping \({\varphi: \mathfrak{A}\rightarrow \mathfrak{B}}\) between models \({\mathfrak{A}}\) and \({\mathfrak{B}}\) of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area and sharing a side into triangles that have the same property, must be a hyperbolic motion onto \({\varphi(\mathfrak{A})}\). The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for “the triangles abc and abd have the same area.”