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.”
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Pambuccian, V. Mappings preserving the area equality of hyperbolic triangles are motions. Arch. Math. 95, 293–300 (2010). https://doi.org/10.1007/s00013-010-0156-7
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DOI: https://doi.org/10.1007/s00013-010-0156-7