A function $f:\mathbb{R}^2\to \mathbb{R}$ has the property $(\mathcal{P})$ if:
For any finitely many points $A_1,A_2, \dots, A_n$ such that $f(A_1)=f(A_2)=\dots=f(A_n)$, it follows that $A_1,A_2, \dots ,A_n$ form a convex polygon.
Let $Q\in \mathbb{C}[X]$. Prove that the function $f(x,y)=|Q(x+iy)|$ has the propery $(\mathcal{P})$ if and only if all roots of $Q$ are equal.
I tried to think of this geometrically in 3 dimensions, but I honestly don't even know how to approach it. If we take $n$ complex numbers wich do not form a convex polygon, then we are assured that not all $|Q(z_1)|,\dots ,|Q(z_n)|$ are equal. I don't know how to proceed.