Solution
= Solution
Let the exterior ball be $B(y,R)$, tangent at $x_0$. Then $|x-y|\geq R$ on $\overline\Omega$, with equality only at $x_0$. Choose $\delta=R^{-1}$ and define
$$
w(x)=\log\frac{|x-y|}{R}.
$$
In two dimensions, $\log|x-y|$ is a <harmonic function> away from $y$, so $\Delta w=0\leq0$ in $\Omega$. Moreover $w(x_0)=0$ and $w>0$ on $\overline\Omega\setminus\{x_0\}$. Thus $w$ is a <barrier for the Dirichlet problem>, and $\boxed{x_0\text{ is a regular boundary point}}$.