I'm finding it difficult to formulate my intuition for the following problem:
Let $\phi: \mathbb{R}^n \to \mathbb{R}^n$ be an affine function such that $\phi(B_1) = B_2$ for two balls in $\mathbb{R}^n$.
Prove/disprove: $\phi$ preserves angles.
My intuition is that it must preserve angles since otherwise the image of a ball would not be a ball (e.g. some ellipse instead).
I am not sure how to turn this into anything formal. I thought of using the definition of an angle between 3 points $x_1,x_2,x_3$: $cos(\theta) = \frac{(x_3-x_2)\cdot (x_1-x_2)}{|x_3-x_2||x_1-x_2|}$ but I cannot connect this to the two balls and the fact that $\phi$ is affine.
I also thought of first showing that it preserves angles between the two balls and somehow generalizing to anything outside those balls, but this again leaves me stuck with the aforementioned issue.
As a final desperate attempt, if $\phi$ were distance preserving then it would also be angle preserving, but it is not necessarily distance preserving since $(x,y) \mapsto (2x,2y)$ is an affine function which maps $B(0,1)$ to $B(0,2)$ but is not distance preserving.
What am I missing?