Mappings preserving the area equality of hyperbolic triangles are motions

It is shown that a mapping φ: U → B between models U and 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 φ(U). 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."