Find all functions f:\mathbb{R}_+ \to \mathbb{R} such that for any positive reals x, y holds:
f(\sqrt{\frac{x^2+xy+y^2}{3}}) = \frac{f(x)+f(y)}{2}.

I tried to do the usual stuff, plugging 0, 1, but to no avail other than f(x)=a for any constant 'a' being an example. Intuition tells me it's the only function with these properties. Help will be appreciated!