The axiomatizations of plane projective and affine geometry include the axiom of non-collinearity, i.e., of the existence of at least three non-collinear points. It is shown that this axiom, when converted into a suitable rule, is conservative over the other rules in the following sense: if an atomic formula is derivable by all the rules from a given finite number of atomic formulas used as assumptions, it is derivable without the rule of non-collinearity. (Thus, a proper use of existential axioms requires existential conclusions.) By the subterm property for the rules with non-collinearity excluded, derivability by the rules of projective and affine geometry is decidable.

As an immediate application of the decision method, we conclude that any finite set of atomic formulas is consistent. As a second application, we prove the independence of the parallel postulate in affine geometry: a very short proof search is exhaustive but fails to give a derivation. Thus, we see, within the system of geometry, that no derivation can lead to the parallel postulate.

It should be noted that the solution to the decision problem for projective and affine geometries applies only to derivations by the geometric rules. When logical rules are applied, to conclude logically compound formulas, the decision problem is known to have, by a result announced first in Tarski (1949), a negative solution. Finally, it should be noted that the decision methods presented here are provably terminating algorithms of proof search.