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

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