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\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}```

\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}

[![enter image description here][1]][1]

\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}```


  [1]: https://i.sstatic.net/fzb9IVW6.jpg

\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}```

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

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

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?

\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}```


  [1]: https://i.sstatic.net/65uy5p7B.png

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:

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?

\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}```


  [1]: https://i.sstatic.net/fzb9IVW6.jpg