Here is the question;
Prove that there does not exist any nonconstant function of pairs of distinct points $P,Q\in\mathbb{R}^2$ or of triples of distinct non collinear points $P,Q,R\in\mathbb{R}^2$ that is invariant under affine transformations.
So I have started by thinking about the map, call it $f$. We have to show that for any $f$ and any affine transformation $A\textbf{x}+\textbf{b}$ that $A(f(P,Q))+\textbf{b}=f(P,Q)$ I am confused though the map $f$ is not defined and the dimensions might not even make sense.
I think it is something to do with the fact if we have a constant function, we can send all points to the same point and then choose an affine map that fixes this point which may lead to a contradiction? Although really do not know how to go about this problem rigorously.
An affine map in the plane can map any non-collinear triple to any other non-collinear triple. So the only way a function can remain invariant under such a map is by being constant.