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The Centroid: Medians and Isotomic Conjugation

The median configuration and Ceva’s isotomic extension

The completed median and isotomic-conjugation figures for The Centroid through Ceva and Vectors, with complete source.

Methods Ceva's theorem · Vectors · Affine geometry · Isotomic conjugation

Drawing tool: tkz-euclideSource available

Completed figures

A scalene triangle ABC with L, M, and N at the side midpoints. The medians AL, BM, and CN meet at G.
Figure 1. The three medians meet at G.
A scalene triangle ABC with cevian feet D, E, and F defined by internal division ratios. Reversing those ratios gives D-prime, E-prime, and F-prime, whose cevians meet at Q.
Figure 2. The cevians defined by three internal division ratios meet at P, and the reciprocal-ratio cevians meet at Q, the isotomic conjugate of P.

Complete source

Median configuration

% Figure for a Classical Theorem Note
% Article: The Centroid through Ceva and Vectors
% Site: Elementary Geometry
% Website: https://elementarygeometry.org/
% Drawing tool: tkz-euclide
% Output format: SVG
% Last verified: 2026-08-24
%
% Editable source for public/diagrams/centroid-medians.svg.
% Prepared for the web edition of this note.
% The web build does not compile TeX; export and review the SVG separately.
\documentclass[tikz,border=8pt]{standalone}
\usepackage{tkz-euclide}

\definecolor{egNavy}{HTML}{17243D}
\definecolor{egGray}{HTML}{777671}
\definecolor{egRust}{HTML}{92513F}
\newcommand{\egUseStandardPlateAt}[1]{%
  \pgfresetboundingbox
  \path[use as bounding box]
    ([xshift=-102pt,yshift=-80pt]#1)
    rectangle
    ([xshift=102pt,yshift=80pt]#1);%
}
\newcommand{\egUseStandardPlate}{%
  \coordinate (egPlateCenter) at (current bounding box.center);%
  \egUseStandardPlateAt{egPlateCenter}%
}
\tikzset{
  eg figure/.style={line cap=round,line join=round,every node/.append style={font=\normalsize,text=egNavy}},
  eg main/.style={draw=egNavy,line width=0.75pt},
  eg support/.style={draw=egGray,line width=0.52pt},
  eg support dashed/.style={draw=egGray,line width=0.52pt,dash pattern=on 3pt off 3pt},
  eg accent/.style={draw=egRust,line width=0.75pt},
  eg mark/.style={draw=egNavy,line width=0.50pt}
}

\begin{document}
\begin{tikzpicture}[eg figure,scale=1.0]
  \tkzDefPoint(-0.5,4.2){A}
  \tkzDefPoint(-3,0){B}
  \tkzDefPoint(3,0){C}

  \tkzDefMidPoint(B,C)\tkzGetPoint{L}
  \tkzDefMidPoint(C,A)\tkzGetPoint{M}
  \tkzDefMidPoint(A,B)\tkzGetPoint{N}
  \tkzInterLL(A,L)(B,M)\tkzGetPoint{G}

  \tkzDrawSegments[eg support dashed](A,L B,M C,N)
  \tkzDrawSegments[eg main](A,B B,C C,A)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C,L,M,N,G)

  \tkzLabelPoints[above](A)
  \tkzLabelPoints[below left](B)
  \tkzLabelPoints[below right](C)
  \tkzLabelPoints[below](L)
  \tkzLabelPoints[right,yshift=2pt](M)
  \tkzLabelPoints[left,yshift=2pt](N)
  \tkzLabelPoints[below left,xshift=2pt,yshift=-2pt](G)
  \egUseStandardPlate
\end{tikzpicture}
\end{document}

Isotomic-conjugation configuration

% Figure for a Classical Theorem Note
% Article: The Centroid through Ceva and Vectors
% Site: Elementary Geometry
% Website: https://elementarygeometry.org/
% Drawing tool: tkz-euclide
% Output format: SVG
% Last verified: 2026-08-24
%
% Editable source for public/diagrams/centroid-isotomic-conjugation.svg.
% Prepared for the web edition of this note.
% The web build does not compile TeX; export and review the SVG separately.
\documentclass[tikz,border=8pt]{standalone}
\usepackage{tkz-euclide}
\usetikzlibrary{calc}

\definecolor{egNavy}{HTML}{17243D}
\definecolor{egGray}{HTML}{777671}
\definecolor{egRust}{HTML}{92513F}
\newcommand{\egUseStandardPlateAt}[1]{%
  \pgfresetboundingbox
  \path[use as bounding box]
    ([xshift=-102pt,yshift=-80pt]#1)
    rectangle
    ([xshift=102pt,yshift=80pt]#1);%
}
\newcommand{\egUseStandardPlate}{%
  \coordinate (egPlateCenter) at (current bounding box.center);%
  \egUseStandardPlateAt{egPlateCenter}%
}
\tikzset{
  eg figure/.style={line cap=round,line join=round,every node/.append style={font=\normalsize,text=egNavy}},
  eg main/.style={draw=egNavy,line width=0.75pt},
  eg support/.style={draw=egGray,line width=0.52pt},
  eg support dashed/.style={draw=egGray,line width=0.52pt,dash pattern=on 3pt off 3pt},
  eg accent/.style={draw=egRust,line width=0.75pt},
  eg mark/.style={draw=egNavy,line width=0.50pt}
}

\begin{document}
\begin{tikzpicture}[eg figure,scale=1.0]
  % Keep this triangle identical to centroid-medians.tex.
  \tkzDefPoint(-0.5,4.2){A}
  \tkzDefPoint(-3,0){B}
  \tkzDefPoint(3,0){C}

  % BD:DC=3:2, CE:EA=2:1, AF:FB=1:3.
  \coordinate (D) at ($(B)!3/5!(C)$);
  \coordinate (E) at ($(C)!2/3!(A)$);
  \coordinate (F) at ($(A)!1/4!(B)$);

  % Reverse the ratios to obtain the isotomic feet:
  % BD':D'C=2:3, CE':E'A=1:2, AF':F'B=3:1.
  \coordinate (Dprime) at ($(B)!2/5!(C)$);
  \coordinate (Eprime) at ($(C)!1/3!(A)$);
  \coordinate (Fprime) at ($(A)!3/4!(B)$);

  \tkzInterLL(A,D)(B,E)\tkzGetPoint{P}
  \tkzInterLL(A,Dprime)(B,Eprime)\tkzGetPoint{Q}

  \tkzDrawSegments[eg support dashed](A,D B,E C,F)
  \tkzDrawSegments[eg support dashed](A,Dprime B,Eprime C,Fprime)
  \tkzDrawSegments[eg main](A,B B,C C,A)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C,D,E,F,Dprime,Eprime,Fprime,P,Q)

  \tkzLabelPoints[above](A)
  \tkzLabelPoints[below left](B)
  \tkzLabelPoints[below right](C)
  \tkzLabelPoint[below](D){$D$}
  \tkzLabelPoint[below](Dprime){$D'$}
  \tkzLabelPoint[above right,yshift=-1pt](E){$E$}
  \tkzLabelPoint[right,yshift=2pt](Eprime){$E'$}
  \tkzLabelPoint[above left](F){$F$}
  \tkzLabelPoint[left](Fprime){$F'$}
  \tkzLabelPoint[above,xshift=4pt](P){$P$}
  \tkzLabelPoint[below,xshift=-5pt](Q){$Q$}
  \egUseStandardPlate
\end{tikzpicture}
\end{document}

Read the article

Read The Centroid through Ceva and Vectors