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There are several ways to define an affine space, either by starting from a transitive action of a vector space on a set of points, or listing sets of axioms related to parallelism in the spirit of Euclid.

On the other hand, it is often said that affine geometry is the geometry of the barycenter. It should be possible to define an affine space by postulating the existence of an operation called the barycenter, $$ ({\bf R} \times {\cal E}) \times ({\bf R} \times {\cal E}) \mapsto {\cal E} $$ that is homogeneous , associative, etc, without talking about vector spaces or parallelism at first, and then recover the basic geometric notions from here: an affine subspace is the set of all barycenters obtained from a finite set of points and so on.

I have never seen such an axiomatic, yet it seems quite natural. Are there any references that take that path to affine geometry?

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