draw line according to other angle

Question

In below example, how can I draw a line through C and satisfy (current D point should be changed)

\documentclass[border=3mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc,angles,quotes}
    \begin{document}
    \begin{tikzpicture}[scale=2]
    \tikzset{dot/.style={circle,inner sep=1pt,fill,label={#1},name=#1},
    extended line/.style={shorten >=-#1,shorten <=-#1},
    extended line/.default=1cm
    }
    \pgfmathsetmacro\r{1}   
    \coordinate (O) at (-2,1);
    \path (O) -- ++(130:\r) coordinate (A)
          (O) -- ++(30:\r) coordinate (B)
          (O) -- ++(-90:\r) coordinate (C)
          ;
     \coordinate (D) at (0,0);
    \draw (O) circle (\r);
    \draw (A) -- (B) -- (C) -- cycle 
    pic["$\alpha$",draw,angle eccentricity=1.8,angle radius=6] {angle=C--A--B}
    pic["$\beta$",draw,angle eccentricity=1.8,angle radius=6] {angle=D--C--B};
    \draw[red,extended line=2cm] (C) -- (D);

    \foreach \x in {A,B,C,D,O} {
        \node[circle,fill,inner sep=0,minimum size=2pt,label=\x] at (\x) {};
    }
    \end{tikzpicture}
\end{document}

Output:

Answer 1

Your setup already has the property alpha=beta. Here is a revised version that addresses the clarification in the comment.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{calc,angles,quotes}
\begin{document}
\begin{tikzpicture}[scale=2]
    \tikzset{dot/.style={circle,inner sep=1pt,fill,label={#1},name=#1},
    extended line/.style={shorten >=-#1,shorten <=-#1},
    extended line/.default=1cm
    }
    \pgfmathsetmacro\r{1} 
    \coordinate (O) at (-2,1);
    \path (O) -- ++(130:\r) coordinate (A)
          (O) -- ++(30:\r) coordinate (B)
          (O) -- ++(-90:\r) coordinate (C)
          ;
    \path let \p1=($(B)-(A)$),\p2=($(C)-(A)$),
    \p3=($(B)-(C)$),\n1={-atan2(\y1,\x1)+atan2(\y2,\x2)+atan2(\y3,\x3)} 
    in  ($(C)+(\n1:2)$) coordinate (D);
    \draw (O) circle (\r);
    \draw (A) -- (B) -- (C) -- cycle 
    let \p1=($(B)-(A)$),\p2=($(C)-(A)$),
    \p3=($(B)-(C)$),\p4=($(D)-(C)$),
    \n1={atan2(\y1,\x1)-atan2(\y2,\x2)},
    \n2={atan2(\y3,\x3)-atan2(\y4,\x4)} in
    pic["$\alpha=\pgfmathparse{\n1}\pgfmathprintnumber\pgfmathresult$"
    {anchor=160},draw,angle eccentricity=1.8,angle radius=6] {angle=C--A--B}
    pic["$\beta=\pgfmathparse{\n2}\pgfmathprintnumber\pgfmathresult$"
    {anchor=200}
    ,draw,angle eccentricity=1.8,angle radius=6] {angle=D--C--B};
    \draw[red,extended line=2cm] (C) -- (D);

    \foreach \x in {A,B,C,D,O} {
        \node[circle,fill,inner sep=0,minimum size=2pt,label=$\x$] at (\x) {};
    }
    \end{tikzpicture}
\end{document}

An animation:

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{calc,angles,quotes,through}
\tikzset{circle through 3 points/.style n args={3}{% https://tex.stackexchange.com/a/461180
insert path={let    \p1=($(#1)!0.5!(#2)$),
                    \p2=($(#1)!0.5!(#3)$),
                    \p3=($(#1)!0.5!(#2)!1!-90:(#2)$),
                    \p4=($(#1)!0.5!(#3)!1!90:(#3)$),
                    \p5=(intersection of \p1--\p3 and \p2--\p4)
                    in },
at={(\p5)},
circle through= {(#1)}
}}
\begin{document}
    \tikzset{dot/.style={circle,inner sep=1pt,fill,label={#1},name=#1},
    extended line/.style={shorten >=-#1,shorten <=-#1},
    extended line/.default=1cm
    }
\foreach \X in {1,1.1,...,2,1.9,1.8,...,1.1}    
    {\begin{tikzpicture}[scale=2]
    \pgfmathsetmacro\r{1} 
    \path[use as bounding box] (-4,3)  rectangle (1,-0.3); % for animation
    \coordinate (O) at (-2,1);
    \path (O) -- ++(130:\X*\r) coordinate (A)
          (O) -- ++(30:\r) coordinate (B)
          (O) -- ++(-90:\r) coordinate (C)
          ;
    \path let \p1=($(B)-(A)$),\p2=($(C)-(A)$),
    \p3=($(B)-(C)$),\n1={-atan2(\y1,\x1)+atan2(\y2,\x2)+atan2(\y3,\x3)} 
    in  ($(C)+(\n1:2)$) coordinate (D);
    \draw[gray,thin] (O) circle (\r);
    \node[circle through 3 points={A}{B}{C},draw=blue](O'){};
    \draw (A) -- (B) -- (C) -- cycle 
    let \p1=($(B)-(A)$),\p2=($(C)-(A)$),
    \p3=($(B)-(C)$),\p4=($(D)-(C)$),
    \n1={atan2(\y1,\x1)-atan2(\y2,\x2)},
    \n2={atan2(\y3,\x3)-atan2(\y4,\x4)} in
    pic["$\alpha=\pgfmathparse{\n1}\pgfmathprintnumber\pgfmathresult$"
    {anchor=160},draw,angle eccentricity=1.8,angle radius=6] {angle=C--A--B}
    pic["$\beta=\pgfmathparse{\n2}\pgfmathprintnumber\pgfmathresult$"
    {anchor=200}
    ,draw,angle eccentricity=1.8,angle radius=6] {angle=D--C--B};
    \draw[red,extended line=2cm] (C) -- (D);

    \foreach \x in {A,B,C,D,O,O'} {
        \node[circle,fill,inner sep=0,minimum size=2pt,label=$\x$] at (\x) {};
    }
    \end{tikzpicture}}
\end{document}