Let $M\subseteq\mathbb R^3$ be a (nondegenerate) closed/open triangle spanned by $p_0,p_1,p_2\in\mathbb R^3$.
Can we show that $M$ is a $2$-dimensional embedded $C^1$ of $\mathbb R^3$ (possibly with boundary)$^1$? Does it matter whether, we consider $M$ as a closed or open triangle?
My idea is that to define $$U:=\{(u,v)\in[0,\infty)^2:u+v\le1\}$$ (possibly adjusting the inequalities depending on whether we consider a closed or open triangle) and $$\psi(u,v):=p_0+ue_1+ve_2,$$ where $e_i:=p_i-p_0$.
Can we show that $(\Omega,\phi):=(\psi(U),\psi^{-1})$ is a chart for $M$?
(If the claim is true at all, please don't consider a chart different from $\phi$ defined above to show the claim.)
If $d\in\mathbb N$ and $k\in\{1,\ldots,d\}$, then $M$ is called $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary if for all $p\in M$, there is an open subset $\Omega$ of $M$ and a $C^1$-diffeomorphism $\phi$ from $\Omega$ onto an open subset of $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. Any such $(\Omega,\phi)$ is called a chart for $M$.