If length of gradient is constant along level curves then for level curves $\beta$, $\gamma$, exists $c \in \mathbb{R}$ with $\gamma = \beta + cn$

Question

I'm trying to prove some results about geodesics and surfaces given by graphs. Im stuck with the following problem:

Let $f: \mathbb{R}^2 \to \mathbb{R}$ such that $\|\nabla f\|^2$ is constant along level curves, then for any pair $\beta$, $\gamma$ of level curves of $f$, there exists $c \in \mathbb{R}$ such that $\gamma = \beta + cn$, where $n$ is the unit normal of $\beta$.

I have unsuccessfully tried to prove that $\|\beta - \gamma\|^2 = constant$ showing that it's derivative is 0