I heard that for any curve in the plane that can be given parametrically by $\vec{r}(t)=\langle x(t),y(t)\rangle$ for $a\leq t\leq b$ that there exists a constant arc length parametrization, i.e. another parametrization $\hat{\vec{r}}(t)=\langle \hat{x}(t),\hat{y}(t)\rangle$ that satisfies $\hat{x}'(t)^2+\hat{y}'(t)^2=1$ and $\hat{\vec{r}}\big((a,b)\big)=\vec{r}\big((a,b)\big)$.
What is the name of this theorem and who proved it? I've been looking around the internet and it seems like the Gauss-Bonnet Theorem comes up a lot, but I don't see the connection between that and this? Maybe I just don't understand it as well as I need to.
The earliest use is due to Leonard Euler in 1775: Methodus facilis omnia Symptomata linearum Curvarum non in eodem Plano sitarum Investigandi, Acta. Acad. Scient. Petropolitanse 1782 I, 1786, 19--57.