What is plane equiaffine geometry?

We provide ∀∃-axiom systems for two variants of plane equiaffine geometry, one whose automorphisms are area preserving affinities, the other whose automorphisms are oriented area preserving affinities. The axiom systems are formulated in first order languages with points as the only individual variables, and a single ternary primitive notion, standing for 'triangle of fixed (oriented or non-oriented) area'. The theorem of G. Martin on area preserving bijections of the plane is seen in a new light.