How do you rotate paths made using the [use path=\name] option in TikZ?

Question

How do you duplicate a path in tikz and rotate the duplicate about its midpoint?

I want to make a copy of the line inside the circle (imaged below with MWE) and draw it through the original line perpendicularly (or any angle I want for that matter).

Thank you very much for your assistance, Pseudonym123

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}[scale=0.3]
\path[name path=circle,draw] (0,0) circle [radius=5cm];
\path[name path=line127] (3,0) -- ++(127:10);
\path[name intersections={of=circle and line127}]
    (intersection-1);
\path[save path=\name](3,0) -- (intersection-1);
\path[draw,rotate=90,use path=\name];%<--- this part isn't rotating
\end{tikzpicture}
\end{document}

Answer 1

If all you need is to rotate straight lines between coordinates (not nodes), I suggest a custom to path with the real start points calculated. That way you can still place nodes along the path.

I put overlay on the invisible line127 so that the bounding box doesn't get updated and named the intersection via a \tikzset since there's no need to use a \path.

Code

\documentclass[tikz]{standalone}
\usetikzlibrary{intersections, calc}
\tikzset{
  rotate straight line/.default=90,
  rotate straight line/.style={
    to path={
      coordinate (@middle@) at ($(\tikztostart)!.5!(\tikztotarget)$)
      ($(@middle@)!1!#1:(\tikztostart)$)--($(@middle@)!1!#1:(\tikztotarget)$)
      \tikztonodes}}}
\begin{document}
\begin{tikzpicture}[scale=0.3]
\path[name path=circle,  draw]    (0,0) circle [radius=5cm];
\path[name path=line127, overlay] (3,0) -- ++(127:10);
\tikzset{name intersections={of=circle and line127}}
\draw[help lines] (3,0) --                          (intersection-1);
\draw             (3,0) to[rotate straight line]    (intersection-1);
\draw[red]        (3,0) to[rotate straight line=45] (intersection-1);
\end{tikzpicture}
\end{document}

Output

Answer 2

I did not understand the question as Qrrbrbirlbel. With the tkz-euclide package (I'm a beginner), I come up with this.

    \documentclass[border=3mm]{standalone}
    \usepackage{tkz-euclide}
    \begin{document}
    \begin{tikzpicture}[scale=0.3]
        \tkzDefPoints{0/0/P,3/0/A}
        \coordinate (B) at ($(A)+(127:10)$);% we can use tikz
    %       (AB)  is  your line127
    \tkzDefCircle[R](P,5) \tkzGetPoint{C}
    \tkzDrawCircles(P,C)

    \tkzInterLC[near](A,B)(P,C) \tkzGetPoints{d}{D}
    \tkzDrawSegment(A,D)% D is the intersection point

    \tkzDefMidPoint(A,D) \tkzGetPoint{M}
    %\tkzDrawPoints(A,D,M)

    \tkzDefPointBy[rotation in rad= center M angle -pi/2](A) \tkzGetPoint{A'}
    \tkzDefPointBy[rotation in rad= center M angle -pi/2](D) \tkzGetPoint{D'}
    %\tkzDrawPoints(A',D')
    \tkzDrawSegment(A',D')
    %\tkzLabelPoints(A,D,M,A',D')
    \end{tikzpicture}
\end{document}

Answer 3

Here's a solution using the spath3 library (it's recently updated on CTAN so you might need to update your distribution). This library allows you to save a path and then manipulate it, including applying transformations. The transform option on the use key allows you to specify a transformation to be applied to the path before it is re-used, and the spath cs coordinate system allows you to specify a coordinate along a path. So with the rotate around key it's not too hard to do what you want.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/664167/86}
\usepackage{tikz}
\usetikzlibrary{
  intersections,
  spath3
}
\begin{document}
\begin{tikzpicture}[scale=0.3]
\path[name path=circle,draw] (0,0) circle [radius=5cm];
\path[name path=line127] (3,0) -- ++(127:10);
\path[name intersections={of=circle and line127}]
    (intersection-1);
\path[draw,red,spath/save=name](3,0) -- (intersection-1);
\path[draw,blue,
  spath/use={
    name,
    transform={
      rotate around={90:(spath cs:name .5)}
    }
  }
];%<--- this part isn't rotating
\end{tikzpicture}
\end{document}

I've drawn the original path as well so that you can see that it does what you asked for:

Answer 4

This is rather a comment than an answer.

For a path, duplicating/copying, rotating about some point, etc. is usual path operations; and can be done in plain Asymptote with the slogan "let it be as it is", see the simple code below. One can read and understand it even not knowing Asymptote programming before.

The spath3 library enhances TikZ's path operations (see Andrew's answer). Without using this library, the task still can be done in plain TikZ with a bit tricky code. In my opinion, it's not really defective of TikZ; it's just how TikZ simplify a way of drawing.

% https://www.overleaf.com/read/pjqpgyztmwcf
\documentclass{standalone}
\usepackage{asymptote}
\begin{document}
\begin{asy}
// http://asymptote.ualberta.ca/
size(6cm);
path cir=circle((0,0),5);
pair A=(3,0), B=A+10dir(127);
path line1=A--B;
pair C=intersectionpoint(cir,line1); 
pair M=(A+C)/2;
dot(M,blue);
// choose 90 for perpendicular one
real al=90, be=45;    
path line2=rotate(al,M)*(A--C);
path line3=rotate(be,M)*(A--C);
draw(cir);
draw(line1,lightblue+.8pt+opacity(.5));
draw(A--C,blue);
draw(line2,red);
draw(line3,orange);
\end{asy}
\end{document}