Is there a way to show that harmonic functions satisfy the mean-value property by directly using the solution to the Dirichlet problem in a disc?
To be slightly more precise, let $f : U \to \mathbb{R}$ be harmonic, and take a disc $D(a,r) \subset U$. I want to show that $f(a) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(a + re^{i\theta}) \ d\theta$ using only the fact that there exists a solution to the Dirichlet problem in the disc $D(a,r)$, that is, given $g : \partial D(a,r) \to \mathbb{R}$ continuous there exists a function $v : \overline{D(a,r)} \to \mathbb{R}$ continuous and harmonic on the interior such that $g(z) = v(z)$ on $\partial D(a,r)$.
Can this be done and, if so, how? All of the proofs I've found use some version of Green's theorem which I would like to avoid, if possible.