tikz / tikzlibrary{calc,intersections}: draw a tangent line that intersects secant at right angle

Question

I need to draw a tangent line that intersects a secant line at a right angle outside the circle.

The MWE below correctly renders a circle and a secant line, and labels the intersection points. However, I can't figure out how to reorient the tangent line so it intersects the secant at a 90° angle outside the circle.

Questions:

❓ Questions:

1. Is my secant line and labeling code efficient?
2. How do I modify the code so the tangent intersects the secant at a right angle outside the circle?

Thank you for taking time to assist me.

mwe:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc,intersections}
\usepackage{amsmath}

\begin{document}

\begin{tikzpicture}[scale=1.5]
    % Draw the circle with a name for intersection calculations
    \draw[thick, name path=circle] (0,0) circle (2);
    
    % Mark the center
    \fill (0,0) circle (1.5pt);
    \node at (0.2,-0.2) {$O$};
    
    % Define the secant line path (extend beyond circle)
    \path[name path=secant] (-1,3) -- (3,-2);
    
    % Find intersection points automatically and store them
    \path[name intersections={of=circle and secant}] 
        (intersection-1) coordinate (A)
        (intersection-2) coordinate (B);
    
    % Extend the secant line beyond the circle in both directions
    \draw[thick, red] (-1,3) -- (3,-2);
    
    % Mark intersection points
    \fill[red] (A) circle (1.5pt) (B) circle (1.5pt);
    
    % Labels for intersection points
    \node at (A) [above right] {$A$};
    \node at (B) [below left] {$B$};
    
    % Define tangent point on the circle
    \coordinate (T) at (1.6, 1.2);  % Point on circle where tangent touches
    
    % Draw tangent line (perpendicular to radius at T)
    % The radius to T has direction (1.6, 1.2), so perpendicular direction is (-1.2, 1.6)
    \draw[thick, blue] ($(T) + 2*(-1.2, 1.6)$) -- ($(T) + 2*(1.2, -1.6)$);
    
    % Mark the tangent point
    \fill[blue] (T) circle (1.5pt);
    \node at (T) [above left] {$T$};
    
\end{tikzpicture}

\end{document}

Answer 1

You could utilize polar coordinates like this:

\documentclass[
    tikz
    ,border = 1cm
]{standalone}
\begin{document}
    \foreach \thedegree in {5,10,...,360} {
        \begin{tikzpicture}
            \useasboundingbox (-2,-2) rectangle (2,2);
            \draw (0,0) circle[radius = 1];
            \draw 
                (0,0) -- (\thedegree:{cos(\thedegree)})
                (\thedegree:{cos(\thedegree)}) -- ++(\thedegree+90:-2) 
                (\thedegree:{cos(\thedegree)}) -- ++(\thedegree+90:2)
                (0,0) -- (\thedegree+90:1)
                (\thedegree+90:1) -- ++(\thedegree:-2)
                (\thedegree+90:1) -- ++(\thedegree:2)
            ;
        \end{tikzpicture}
    }
\end{document}