Let $A$ be an affine space associated to a vector space $V$ over a field $k$ (i.e. $V$ acts as an additive group on the set $A$ and this action is free and transitive). I want to prove that if a nonempty subset $S$ of $A$ contains all points of the form $p+\lambda \vec{pq}\in S$ for all $p,q\in S$ and $\lambda\in k$, then $S$ is an affine subspace of $A$ (i.e., $S=p+W$ for some $p\in A$ and some vector subspace $W$ of $V$).
My idea was to put $$W=\{\vec{qq'}|q,q'\in S\}$$ and prove that $W$ is a vector subspace of $V$, but i'm stuck on proving that it is closed under addition.
Our hypothesis about the subset $S\subset A$ is $$\forall p,q\in S\quad(pq)\subset S,$$ and we want to derive that $$\forall a,b,c,d\in S\quad\exists p,q\in S\quad\overrightarrow{ab}+\overrightarrow{cd}=\overrightarrow{pq}.$$
We shall assume that the characteristic is $\ne2$ (when $k=\Bbb Z_2,$ the hypothesis is satisfied by every $S\subset A$).
We shall prove that we can even choose $p=a,$ i.e. that $$q:=a+\overrightarrow{ab}+\overrightarrow{cd}=b+\overrightarrow{cd}\in S.$$
Let $$r:=b+\frac12\overrightarrow{bd}\in S.$$ Then, $$b+\overrightarrow{cd}=c+\overrightarrow{cb}+\overrightarrow{cd}=c+2\overrightarrow{cr}\in S.$$