Functions proof. Find all functions $$f: \mathbb{Z} \rightarrow \mathbb{Z}$$ such that
$$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$
for all integers $$a, b, c$$ satisfying $$a+b+c=0$$
I have no idea how to even begin this one? Any comments?

I think using $f(x)=-x$ works, since $0=(-a-b-c)²=a²+b²+c²-2ab-2bc-2ac$, so $f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$
