4

I have two concentric circles centered at O drawn. I specify that the radius of the smaller circle is 1. The radius of the bigger circle is determined so that an inscribed angle LMN has measure 20 degrees, is bisected by radius OM, and is tangent to the smaller circle at S and T so that angles SOM and TOM have measure 80 degrees.

My concern is chord MN is not drawn. Chord LM is draw but not chord MN. Here are the calculations showing the lengths of the congruent chords LM and MN.

`triangle{OSM}` and `triangle{OTM}` are congruent, right triangles.
Since `OM` bisects `angle{LMN}`, and the measure of `angle{LMN}` is 20 degrees,
`angle{LMO}` and `angle{NMO` both have measure `10` degrees.

r is the radius of the smaller circle.
According to the Law of Sines, |OM| = r/sin(10). By the Pythagorean Theorem,
|MS| = |MT| = (r/sin(10))\sqrt{1 - sin^{2}(10)} =(r/sin(10))cos(10) = r*cot(10).

I declare r=1. So, the radius of the bigger circle is cot(10). With the following commands, I expected to have angle{LMN} to be an inscribed angle in the bigger triangle.

\coordinate (M) at ({-cot(10)},0);
\coordinate (L) at ($(M) +(10:{2*cot(10)})$);
\coordinate (N) at ($(M) +(-10:{2*cot(10)})$);

That didn't happen, LM and MN are drawn a bit past the circle at M and N. How does this happen?

I used the following commands to compensate for the mistakes from the calc package.

\draw[name path=bigger_circle] (O) circle ({cot(10)});
\path[name path=chord_LM] (M) -- (L);
\draw[name path=chord_MN] (M) -- (N);
\coordinate[name intersections={of=bigger_circle and chord_LM, by=corrected_location_for_L}];
\coordinate[name intersections={of=bigger_circle and chord_MN, by=corrected_location_for_N}];

Chord LM is drawn, but N seems to be located at M, and so chord MN is not drawn.

\documentclass{amsart}
\usepackage{tikz}

\usetikzlibrary{calc,intersections}


\begin{document}

\begin{tikzpicture}

%Two concentric circles are drawn.
%
\coordinate (O) at (0,0);
\draw[fill] (O) circle (1.5pt);
\draw (O) circle (1);
\draw[name path=bigger_circle] (O) circle ({cot(10)});
%

%
\coordinate (S) at (100:1);
\draw[fill] (S) circle (1.5pt);
\coordinate (T) at (-100:1);
\draw[fill] (T) circle (1.5pt);
%
\coordinate (M) at ({-cot(10)},0);
%
\coordinate (L) at ($(M) +(10:{2*cot(10)})$);
\coordinate (N) at ($(M) +(-10:{2*cot(10)})$);
%
\path[name path=chord_LM] (M) -- (L);
\path[name path=chord_MN] (M) -- (N);
%
%The calc package is drawing the chords LM and MN too long. So, the intersections package is used.
%
\coordinate[name intersections={of=bigger_circle and chord_LM, by=corrected_location_for_L}];
\coordinate[name intersections={of=bigger_circle and chord_MN, by=corrected_location_for_N}];
%
\draw (M) -- (corrected_location_for_L);
\draw[green] (M) -- (corrected_location_for_N);


%The labels for the points are typeset.
\path node[anchor=west, inner sep=0, font=\footnotesize] at ($(O) +(0.15,0)$){$O$};
\path node[anchor=east, inner sep=0, font=\footnotesize] at ($(M) +(-0.15,0)$){$M$};
\path let \p1=($(L)-(M)$), \n1={atan(\y1/\x1)} in node[anchor={\n1+180}, inner sep=0, font=\footnotesize] at ($(corrected_location_for_L) +({\n1}:0.15)$){$L$};
\path let \p1=($(M)-(N)$), \n1={atan(\y1/\x1)} in node[anchor={\n1+180}, inner sep=0, font=\footnotesize] at ($(corrected_location_for_N) +({\n1}:0.15)$){$N$};
\path node[anchor={80-180}, inner sep=0, font=\footnotesize] at ($(S) +(80:0.15)$){$S$};
\path node[anchor={-80+180}, inner sep=0, font=\footnotesize] at ($(T) +(-80:0.15)$){$T$};

\draw[fill=green] (corrected_location_for_L) circle (2pt);
\draw[fill=green] (corrected_location_for_N) circle (2pt);


\end{tikzpicture}

\end{document}
  • You state that the radius of the bigger circle is 1/sin(10), but you have \draw[name path=bigger_circle] (O) circle ({cot(10)}). So, your bigger circle is a bit small. – user74973 Jan 26 '17 at 17:24
  • Draw the bigger circle with \draw (O) circle ({1/sin(10)}). Locate M with \coordinate (M) at ({-1/sin(10)},0). – user74973 Jan 26 '17 at 17:25
4

You have picked up a wrong intersection point, each line meets the circle at the desired point and at M. To be absolutely certain which point you pick up you should use the sort by option for the intersections. In your case using the order along the chord is most appropriate. E.g. for the first set of intersections you can write

\coordinate[name intersections={of=bigger_circle and chord_ML,
by={tmp,corrected_location_for_L}, sort by=chord_ML}];

sorting the intersections by their order along chord_ML and labelling them by tmp and corrected_location_for_L.

Sample output

\documentclass{amsart}

\usepackage{tikz}

\usetikzlibrary{calc,intersections}

\begin{document}

\begin{tikzpicture}

%Two concentric circles are drawn.
%
\coordinate (O) at (0,0);
\draw[fill] (O) circle (1.5pt);
\draw (O) circle (1);
\draw[name path=bigger_circle] (O) circle ({cot(10)});
%

%
\coordinate (S) at (100:1);
\draw[fill] (S) circle (1.5pt);
\coordinate (T) at (-100:1);
\draw[fill] (T) circle (1.5pt);
%
\coordinate (M) at ({-cot(10)},0);
%
\coordinate (L) at ($(M) + (10:{2*cot(10)})$);
\coordinate (N) at ($(M) + (-10:{2*cot(10)})$);
%
\path[name path=chord_ML] (M) -- (L);
\path[name path=chord_MN] (M) -- (N);
%
%The calc package is drawing the chords LM and MN too long. So, the intersections package is used.
%
\coordinate[name intersections={of=bigger_circle and chord_ML,
by={tmp,corrected_location_for_L}, sort by=chord_ML}];
\coordinate[name intersections={of=bigger_circle and chord_MN,
by={tmp,corrected_location_for_N}, sort by=chord_MN}];
%
\draw (M) -- (corrected_location_for_L);
\draw[green] (M) -- (corrected_location_for_N);

%The labels for the points are typeset.
\path node[anchor=west, inner sep=0, font=\footnotesize] at ($(O) +(0.15,0)$){$O$};
\path node[anchor=east, inner sep=0, font=\footnotesize] at ($(M) +(-0.15,0)$){$M$};
\path let \p1=($(L)-(M)$), \n1={atan(\y1/\x1)} in node[anchor={\n1+180}, inner sep=0, font=\footnotesize] at ($(corrected_location_for_L) +({\n1}:0.15)$){$L$};
\path let \p1=($(M)-(N)$), \n1={atan(\y1/\x1)} in node[anchor={\n1+180}, inner sep=0, font=\footnotesize] at ($(corrected_location_for_N) +({\n1}:0.15)$){$N$};
\path node[anchor={80-180}, inner sep=0, font=\footnotesize] at ($(S) +(80:0.15)$){$S$};
\path node[anchor={-80+180}, inner sep=0, font=\footnotesize] at ($(T) +(-80:0.15)$){$T$};

\draw[fill=green] (corrected_location_for_L) circle (2pt);
\draw[fill=green] (corrected_location_for_N) circle (2pt);

\end{tikzpicture}

\end{document}
  • By removing corrected_location_for_N from the command that locates the intersections of MN with the bigger circle and including the command \coordinate (corrected_location_for_N) at (intersection-2), chord MN is drawn. – user74973 Jan 26 '17 at 17:18
  • The commands for chord MN has it intersecting the circle twice - one of them at M. TikZ labels these intersections intersection-1 and intersection-2. – user74973 Jan 26 '17 at 17:18
  • 1
    @user74973 That is a usable alternative. – Andrew Swann Jan 26 '17 at 17:24
1

I don't know about the length of the line, but the wrong label location is because you're using the wrong intersection.

There are two intersections between the circle and the chords. One is at M, the other at L/N. When you use by={a}, the first intersection TikZ finds is given the name {a}, if you use by={a,b} the first gets a and the second b.

Exactly why you get the correct intersection in one case but not the other, I cannot say for sure, but it will be due to how TikZ finds the intersections. For example, assuming it looks along the circle, starting at zero degrees and working its way counterclockwise, for ML the first intersection it finds is at L, and the second at M. For MN, the first intersection it finds will be at M.

So, below is a complete code that does what you're after. I did also change some other things, mainly in how the labels are drawn, but you can ignore that if you prefer your own method of course.

enter image description here

\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,intersections}
\begin{document}

\begin{tikzpicture}[every label/.append style={font=\footnotesize}]
%Two concentric circles are drawn.
%
\coordinate [label=right:{$O$}] (O) at (0,0);
\draw (O) circle (1);
\draw[name path=bigger_circle] (O) circle[radius={cot(10)}];
%

%
\coordinate [label=above:{$S$}] (S) at (100:1);
\coordinate [label=below:{$T$}] (T) at (-100:1);

%
\foreach \x in {O,S,T}
  \draw[fill] (\x) circle (1.5pt);


\coordinate [label=left:{$M$}] (M) at ({-cot(10)},0);
%
\coordinate (L) at ($(M) +(10:{2*cot(10)})$);
\coordinate (N) at ($(M) +(-10:{2*cot(10)})$);
%
\path[name path=chord_LM] (M) -- (L);
\path[name path=chord_MN] (M) -- (N);
%
%The calc package is drawing the chords LM and MN too long. So, the intersections package is used.
%
\path[name intersections={of=bigger_circle and chord_LM, by={corrected_location_for_L,i}}];
\path[name intersections={of=bigger_circle and chord_MN, by={i,corrected_location_for_N}}];
%
\draw (M) -- (corrected_location_for_L);
\draw[green] (M) -- (corrected_location_for_N);

\draw[fill=green] (corrected_location_for_L) circle (2pt);
\draw[fill=green] (corrected_location_for_N) circle (2pt);

\node [font=\footnotesize] at ($(L)!-2mm!(M)$)  {$L$};
\node [font=\footnotesize] at ($(N)!-2mm!(M)$)  {$N$};
\end{tikzpicture}
\end{document}

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