Given an affine space $A$, we can formally generate a vector space $V$ by points of $A$, subject to the affine relations among them found in $A$. In particular, if $a$ and $b$ are points of $A$, then $a + b$ will be defined as an element of $V$, and the middle point of the segment $[ab]$ will be $\frac{a}{2} + \frac{b}{2}$ with respect to the vector operations of $V$.
Does this construction have a standard name? What has it been used for?
Update.
The construction I am interested in seems to be just a different formulation of the Barycentric Calculus of Möbius, see
Möbius, August Ferdinand (1827). Der barycentrische Calkul.
Allardice, R. (1891). The Barycentric Calculus of Möbius. Proceedings of the Edinburgh Mathematical Society, 10, 2-21.