Rotate a path around a point

You are overwriting the nodes so it's a little trickier but essentially you can scope the path and use rotate around transformation.

\documentclass[tikz]{standalone}

\begin{document}
\begin{tikzpicture}
    \providecommand* \angle {30}
    \coordinate[label=above left:A](A) at (2,3);
    \coordinate[label=below:B](B) at (0,0);
    \coordinate[label=below:C](C) at (6,0);
    \draw
        (A) -- (B) -- (C) -- cycle;

\begin{scope}[rotate around={60:(A)}]
    \coordinate[label=above left:A](A) at (2,3);
    \coordinate[label=below:B](B) at (0,0);
    \coordinate[label=below:C](C) at (6,0);
    \draw[dashed,red]
        (A) -- (B) -- (C) -- cycle;
\end{scope}

\end{tikzpicture}
\end{document}