[Tex/LaTex] Drawing a tangent from a point outside of a circle to it!

circlestikz-pgf

I have an easy question. I want to draw the tangents from origin to a circle with center at e.g. (0,2) and radius 1.5. Here the code of the circle:

\begin{tikzpicture}
\draw[->] (0,-.5) -- (3,-.5) node[right] {$x$}; 
\draw[->] (0,-.5) -- (0,2) node[above] {$y$};
\draw (0,2) circle (1.5);
\end{tikzpicture}

I need a simple solution, because I'm not good at drawing with tikz.
Thanks!

Best Answer

Tangent coordinate system

TikZ knows a tangent coordinate system for shapes, if library calc is loaded, see section "13.2.4 Tangent Coordinate Systems" of the PGF/TikZ manual.

The circle is drawn as node with circular shape:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
\begin{tikzpicture}
\draw[->] (0,-.5) -- (3,-.5) node[right] {$x$};
\draw[->] (0,-.5) -- (0,2) node[above] {$y$};
% \draw (0,2) circle (1.5);
\node[circle, draw] (c) at (0, 2) [minimum size=3cm] {};
\draw[red]
  (0, 0) coordinate (a)
  -- (tangent cs:node=c, point={(a)}, solution=1)
  (0, 0)
  -- (tangent cs:node=c, point={(a)}, solution=2)
;
\end{tikzpicture}
\end{document}

Trigonometry

The angle and length of the lines can be calculated using the law of sines and the Pythagorean theorem:

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
\draw[->] (0,-.5) -- (3,-.5) node[right] {$x$};
\draw[->] (0,-.5) -- (0,2) node[above] {$y$};
\draw (0,2) circle (1.5);
\pgfmathsetmacro\angle{asin(1.5/2)}
\pgfmathsetmacro\len{sqrt(2*2 - 1.5*1.5)}
\draw[red]
  (0, 0) -- (90 - \angle:\len)
  (0, 0) -- (90 + \angle:\len)
;
\end{tikzpicture}
\end{document}

Intersections

The question from the comment can be solved with library intersections. A line is defined, which goes through the circle. Then the intersections are calculated and the line drawn:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{intersections}
\begin{document}
\begin{tikzpicture}
\draw[->] (0,-.5) -- (3,-.5) node[right] {$x$};
\draw[->] (0,-.5) -- (0,2) node[above] {$y$};
\draw[name path=circle] (0,2) coordinate (center) circle (1.5);
\coordinate (arbitrary point) at ($(center) + (-80:1.5)$);
% PGF/TikZ manual: 13.5.4 The Syntax of Distance Modifiers
\path[name path=line] (0, 0) -- ($(arbitrary point)!3cm!180:(0, 0)$);
\draw[red, name intersections={of=circle and line}]
  (0, 0) -- (intersection-1);
\end{tikzpicture}
\end{document}

Related Solutions

[Tex/LaTex] Tangents to a circle from a point outside of it (tikz)

In the pgf manual page 130 (ver. 3.10), you have the exact example. I just copy it from there for convenience in case anybody needs. It requires \usetikzlibrary{calc}

\begin{tikzpicture}
\draw[help lines] (0,0) grid (3,2);
\coordinate (a) at (3,2);
\node [circle,draw] (c) at (1,1) [minimum size=40pt] {$c$};
\draw[red] (a) -- (tangent cs:node=c,point={(a)},solution=1) --
(c.center) -- (tangent cs:node=c,point={(a)},solution=2) -- cycle;
\end{tikzpicture}

[Tex/LaTex] TikZ: Drawing an arc from an intersection to an intersection

A solution which allows to draw intersection segments of any two intersections is available as tikz library fillbetween.

This library works as general purpose tikz library, but it is shipped with pgfplots and you need to load pgfplots in order to make it work:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}];

\end{tikzpicture}

\end{document}

enter image description here

The key intersection segments is described in all detail in the pgfplots reference manual section "5.6.6 Intersection Segment Recombination"; the key idea in this case is to

create a temporary path temp1 which is the first intersection segment of red and blue, more precisely, it is the first intersection segment in the Left argument in red and blue : red. This path is drawn as thin black path. Substitute its \draw statement by \path to make it invisible.
Compute the desired intersection segment by intersecting temp1 and green and use the correct intersection segment. By trial and error I figured that it is the third segment of path temp1 which is written as L3 (L = left argument in temp1 and green and 3 means third segment of that path).

The argument involves some trial and error because fillbetween is unaware of the fact that end and startpoint are connected -- and we as end users do not see start and end point.

Note that you can connect these path segments with other paths. If such an intersection segment should be the continuation of another path, use -- as before the first argument in sequence. This allows to fill paths segments:

\documentclass{standalone}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{fillbetween}

\begin{document}

\begin{tikzpicture}

\draw [name path=red,red] (120:1.06) circle (1.9);
%\draw [name path=yellow,yellow] (0:1.06) circle (2.12);
\draw [name path=green,green!50!black] (0:0.77) circle (2.41);
\draw [name path=blue,blue] (0:0) circle (1.06);

% substitute this temp path by `\path` to make it invisible:
\draw[name path=temp1, intersection segments={of=red and blue,sequence=L1}];
\draw[red,fill=blue,-stealth,ultra thick, intersection segments={of=temp1 and green,sequence=L3}]  
    [intersection segments={of=temp1 and green, sequence={--R2}}]
;

\end{tikzpicture}

\end{document}