Example: Nine points circle of a triangle

Published 2010-03-17 | Author: Arnaud Lefebvre
In any triangle ABC, there exists a circle passing through nine special points :
  • the 3 middles of sides
  • the 3 vertices projections on opposite sides
  • the middles of segments [HA], [HB] and [HC] such that H is the intersection point of the 3 altitudes

This TikZ code shows these points.

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Nine points circle of a triangle

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% Nine points circle of a triangle
% Author : Arnaud Lefebvre (IREM Rouen)
%
% In any triangle ABC, there exists a circle passing through
% nine special points : 
%  - the 3 middles of sides
%  - the 3 vertices projections on opposite sides
%  - the middles of segments [HA], [HB] and [HC] such that
%    H is the intersection point of the 3 altitudes

% Requirements : 
%  - tkz-2d.sty  
%  - tkz-arith.sty  
%  - tkz-base.sty  
%  - tkz-berge.sty 
%  - developement version of PGF

\documentclass{article}
\usepackage{tikz,tkz-2d}
\usetikzlibrary{calc,through,intersections}
\usepackage{ifthen}
\usepackage{geometry}
\geometry{vmargin=1cm}

% Definition of the command showing the nine points
\newcommand\ninepointscircle[3]{%
\begin{tikzpicture}
	%----------------------------------------------------------
	% Vertices of the triangle
	%----------------------------------------------------------
	\coordinate[label=above:$A$] (A) at (#1);
	\coordinate[label=left:$B$]  (B) at (#2);
	\coordinate[label=right:$C$] (C) at (#3);

	%----------------------------------------------------------
	% Middles of sides
	%----------------------------------------------------------
	\coordinate[label=below:$A'$] (A') at ($(B)!.5!(C)$);
	\coordinate[label=right:$B'$] (B') at ($(A)!.5!(C)$);
	\coordinate[label=left:$C'$]  (C') at ($(B)!.5!(A)$);

	%----------------------------------------------------------
	% Projections of vertices on opposite sides
	%----------------------------------------------------------
	\coordinate[label=below:$H_A$] (HA) at ($(B)!(A)!(C)$);
	\coordinate[label=45:$H_B$]    (HB) at ($(A)!(B)!(C)$);
	\coordinate[label=135:$H_C$]   (HC) at ($(B)!(C)!(A)$);

	%----------------------------------------------------------
	% Drawing the triangle, medians and altitudes
	%----------------------------------------------------------
	\draw[name path=triangle] (A) -- (B) -- (C) -- cycle;
	\draw[color=red,name path=ma] (A)--(A');
	\draw[color=red,name path=mb] (B)--(B');
	\draw[color=red,name path=mc] (C)--(C');
	\draw[fill=blue,color=blue,name path=ha] (A) -- (HA);
	\draw[fill=blue,color=blue,name path=hb] (B) -- (HB);
	\draw[fill=blue,color=blue,name path=hc] (C) -- (HC);
	\tkzRightAngle[color=blue](B/HC/C,C/HA/A,A/HB/B)

	%----------------------------------------------------------
	% Drawing perpendicular bisectors
	%----------------------------------------------------------
	\tkzMathLength(A,HA)
	\path [name path=C1] (A') -- ($(A')!\tkzMathLen pt!90:(C)$);
	\tkzMathLength(B,HB)
	\path [name path=C2] (B') -- ($(B')!\tkzMathLen pt!90:(A)$);
	\tkzMathLength(C,HC)
	\path [name path=C3] (C') -- ($(C')!\tkzMathLen pt!90:(B)$);
	\draw[color=green,name intersections={of=triangle and C1,name=ch1,sort by=C1,total=\t}] 
	   (ch1-\t)--(A') coordinate (OA') at (ch1-\t);
	\draw[color=green,name intersections={of=triangle and C2,name=ch2,sort by=C2,total=\t}] 
	   (ch2-\t)--(B') coordinate (OB') at (ch2-\t);
	\draw[color=green,name intersections={of=triangle and C3,name=ch3,sort by=C3,total=\t}] 
	   (ch3-\t)--(C') coordinate (OC') at (ch3-\t);

	%----------------------------------------------------------
	% Drawing angles
	%----------------------------------------------------------
	\tkzRightAngle[color=green](B/C'/OC',C/A'/OA',A/B'/OB')

	%----------------------------------------------------------
	% Drawing in dashed style in case of obtuse angle
	%----------------------------------------------------------
	\tkzMathLength(B,C)
	\ifthenelse{\isundefined{\la}}{\newlength{\la}}{}
	\setlength{\la}{\tkzMathLen pt}
	\setlength{\la}{.01\la}
	\tkzMathLength(A,C)
	\ifthenelse{\isundefined{\lb}}{\newlength{\lb}}{}
	\setlength{\lb}{\tkzMathLen pt}
	\setlength{\lb}{.01\lb}
	\tkzMathLength(A,B)
	\ifthenelse{\isundefined{\lc}}{\newlength{\lc}}{}
	\setlength{\lc}{\tkzMathLen pt}
	\setlength{\lc}{.01\lc}
	\pgfmathsetmacro{\angle}{acos((\la*\la-\lb*\lb-\lc*\lc)/(-2*\lb*\lc))}
	\pgfmathtruncatemacro\A{\angle}
	\ifthenelse{\A>90}{
	   \draw[style=dashed,color=green] (O)--(OA') (O)--(OB') (O)--(OC'); 
	   \draw [style=dashed, color=black] (A)--(HC) (A)--(HB); 
	   \draw[style=dashed,color=blue] (A)--(H) (HB)--(H) (HC)--(H);}{}
	\pgfmathsetmacro{\angle}{acos((\lb*\lb-\la*\la-\lc*\lc)/(-2*\la*\lc))}
	\pgfmathtruncatemacro\B{\angle}
	\ifthenelse{\B>90}{
	   \draw[style=dashed,color=green] (O)--(OA') (O)--(OB') (O)--(OC'); 
	   \draw [style=dashed, color=black] (B)--(HA) (B)--(HC); 
	   \draw[style=dashed,color=blue] (B)--(H) (HC)--(H) (HA)--(H);}{}
	\pgfmathsetmacro{\angle}{acos((\lc*\lc-\la*\la-\lb*\lb)/(-2*\lb*\la))}
	\pgfmathtruncatemacro\C{\angle}
	\ifthenelse{\C>90}{
	   \draw[style=dashed,color=green] (O)--(OA') (O)--(OB') (O)--(OC'); 
	   \draw [style=dashed, color=black] (C)--(HA) (C)--(HB); 
	   \draw[style=dashed,color=blue] (C)--(H) (HB)--(H) (HA)--(H);}{}

	%----------------------------------------------------------
	% Positioning gravity center, centroid, circumcenter
	% and the nine points circle center
	%----------------------------------------------------------
	\coordinate[label=45:$H$] (H) at (intersection of A--HA and B--HB);
	\coordinate[label=0:$G$] (G) at (intersection of A--A' and B--B');
	\coordinate[label=-135:$O$] (O) at (intersection of OC'--C' and OB'--B');
	\draw[fill=red,color=red] (G) circle (.8pt);
	\draw[fill=blue,color=blue] (H) circle (.8pt);
	\draw[fill=green,color=green] (O) circle (.8pt);
	\coordinate (N) at ($(H)!.5!(O)$);

	%----------------------------------------------------------
	% Drawing the Euler's line (also in case of obtuse angle)
	%----------------------------------------------------------
	\ifthenelse{\la>\lb}{\def\m{max(\la,\lc)}}{\def\m{max(\lb,\lc)}}
	\ifthenelse{\A>90}{\draw[color=orange] (G)--($(G)!1.1!(O)$) (G)--($(G)!1.1!(H)$) node[right]{Euler's line};}{
	\ifthenelse{\B>90}{\draw[color=orange] (G)--($(G)!1.1!(O)$) (G)--($(G)!1.1!(H)$) node[right]{Euler's line};}{
	\ifthenelse{\C>90}{\draw[color=orange] (G)--($(G)!1.1!(O)$) (G)--($(G)!1.1!(H)$) node[right]{Euler's line};}{
	\draw[color=orange] (G)--($(G)!50*\m!(O)$) (G)--($(G)!50*\m!(H)$) node[right]{Euler's line};}}}

	%----------------------------------------------------------
	% Marking points of interest
	%----------------------------------------------------------
	\draw[fill=black] ($(A)!.5!(H)$) circle (.8pt) node[anchor=south west] {$I$};
	\path ($(A)!.5!(H)$) -- node[sloped] {\tiny{//}} (A);
	\path ($(A)!.5!(H)$) -- node[sloped] {\tiny{//}} (H);
	\draw[fill=black] ($(B)!.5!(H)$) circle (.8pt) node[below] {$J$};
	\path ($(B)!.5!(H)$) -- node[sloped] {\tiny{/}} (B);
	\path ($(B)!.5!(H)$) -- node[sloped] {\tiny{/}} (H);
	\draw[fill=black] ($(C)!.5!(H)$) circle (.8pt) node[right] {$K$};
	\path ($(C)!.5!(H)$) -- node[sloped] {\tiny{$\times$}} (C);
	\path ($(C)!.5!(H)$) -- node[sloped] {\tiny{$\times$}} (H);
	\draw[fill=black] (A') circle (.8pt);
	\draw[fill=black] (B') circle (.8pt);
	\draw[fill=black] (C') circle (.8pt);
	\draw[fill=black] (HA) circle (.8pt);
	\draw[fill=black] (HB) circle (.8pt);
	\draw[fill=black] (HC) circle (.8pt);

	%----------------------------------------------------------
	% Drawing the nine points circle
	%----------------------------------------------------------
	\node[fill=lightgray,opacity=.2,draw,circle through=(C'),
	   label=90:$\mathcal{C}$] at (N) {};
\end{tikzpicture}
}

\begin{document}
\pagestyle{empty}

\ninepointscircle{5,6}{0,0}{7,0}\\
\ninepointscircle{10,4}{0,0}{7,0}

\end{document}

Comments

  • #1 Gillian, December 18, 2012 at 7:42 p.m.

    Brilliant!! Thank you! This is incredibly helpful. Gillian

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