We study the classical problem of trade in two-dimensional Euclidean space. It is known that there is no efficient rule for this model that is compatible with dominant strategy incentives, that is, there is no efficient and strategy-proof rule. We observe that, in addition to incentive constraints, informational constraints are also unavoidable for social planners. Thus, we impose the requirement that finite dimensional messages be sufficient information to realize a rule. In addition, we impose the minimal fairness axioms of anonymity and a weakening of non-bossiness, as well as continuity. The result is a class of rules that is similar to those characterized by Barbera and Jackson ["Strategy-proof exchange", Econometrica, 63 (1995), 51-87].