How do we define affine combinations in affine spaces? Suppose we're given a field $A$ and a vector space $X$ over it. Then an affine space over that is a set $P$ (of "points") equipped with an action
$$+ : X \times P \rightarrow P$$
such that


*

*$0_X+p = p$

*$(x+y)+p = x+(y+p)$

*For all $x \in X$, the function $p \in P \mapsto x+p \in P$ is a bijection.


Lets call the whole tuple $(A,X,P)$ an affine system.
Question. Given an affine system $(A,X,P)$ and together with $a,b \in A$ satisfying $a+b=1$, how do we define the affine combination $ap+bq$? Where $p$ and $q$ are elements of $P.$

The third axiom implies there is a map ${-} : P \times P \to X$ such that $p + (q -  p) = q$ for all $p$ and $q$ in $X$. So we may define $a p + b q$ as $p + b (q - p)$.
