From absolute to affine geometry in terms of point-reections, midpoints, and collinearity

We investigate equational theories expressed in terms of the
point-reflection operation $\sigma$ and the midpoint operation $\mu$ that lie strictly between the absolute and the affine theory, proving a number of dependencies and independencies in the process. Several universal theories enlarged with the collinearity predicate also lie strictly between the absolute and the affine theory. The independence models and several proofs were obtained by {\tt Tipi}, an aggregate of automatic theorem provers. To show that no set of equations with at most three variables can axiomatize the affine theory is left as an open problem.