The only operations defined on points in an affine space are
- point-vector addition. this yields a new point.
- point-point subtraction. this yields a vector.
This can be extended to an affine sum, $\sum_i \alpha_i P_i$, where $\sum_i\alpha_i = 1$.
However, it is possible to derive an expression using these operations that should otherwise be undefined.
Imagine points, $P$, $Q$, $R$, and $S$ such that
$ P - Q = R - S $
This leads to
$ P + S = Q + R$
Each side of the equation is now a point-point addition which is not defined. It is also not an affine sum.
How should each side of this equation be interpreted?