This is a step at which I am stuck in proving a theorem.
Suppose $\Omega$ is a $\mathcal C^1$ open connected and bounded subset of $\mathbb R^2$. (By $\mathcal C^1$, I mean that the boundary is defined by a $\mathcal C^1$ function). Let $\nu(x)$ be the outward normal at a boundary point $x \in \partial \Omega$. \Is it possible that $\langle x, \nu(x) \rangle = 0$ for every every $x \in \partial \Omega$?