The Nine-Point Circle: One Homothety

% Figure for a Classical Theorem Note
% Article: The Nine-Point Circle from the Circumcircle
% Site: Elementary Geometry
% Website: https://elementarygeometry.org/
% Drawing tool: tkz-euclide
% Output format: SVG
% Standard plate: 5:4
% Last verified: 2026-09-01
% Accent target: none
%
% Editable source for public/diagrams/nine-point-circle-from-circumcircle.svg.
% Prepared for the web edition of this note.
% The production site serves the committed SVG and does not compile TeX.
\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]
  % Use a uniform similarity of the standard ABC configuration shared by the
  % Euler-line and centroid figures.
  \tkzDefPoint(0,0){S}
  \tkzDefPoint(-0.5,4.2){A0}
  \tkzDefPoint(-3,0){B0}
  \tkzDefPoint(3,0){C0}
  \tkzDefPointBy[homothety=center S ratio 0.75](A0)\tkzGetPoint{A}
  \tkzDefPointBy[homothety=center S ratio 0.75](B0)\tkzGetPoint{B}
  \tkzDefPointBy[homothety=center S ratio 0.75](C0)\tkzGetPoint{C}

  % Circumcenter, orthocenter, and nine-point center.
  \tkzDefTriangleCenter[circum](A,B,C)\tkzGetPoint{O}
  \tkzDefTriangleCenter[ortho](A,B,C)\tkzGetPoint{H}
  \tkzDefMidPoint(O,H)\tkzGetPoint{N}

  % The three side midpoints.
  \tkzDefMidPoint(B,C)\tkzGetPoint{D}
  \tkzDefMidPoint(C,A)\tkzGetPoint{E}
  \tkzDefMidPoint(A,B)\tkzGetPoint{F}

  % The three altitude feet.
  \tkzDefPointBy[projection=onto B--C](A)\tkzGetPoint{X}
  \tkzDefPointBy[projection=onto C--A](B)\tkzGetPoint{Y}
  \tkzDefPointBy[projection=onto A--B](C)\tkzGetPoint{Z}

  % The three midpoints between the vertices and the orthocenter.
  \tkzDefMidPoint(A,H)\tkzGetPoint{P}
  \tkzDefMidPoint(B,H)\tkzGetPoint{Q}
  \tkzDefMidPoint(C,H)\tkzGetPoint{R}

  \tkzDrawCircle[eg support](O,A)
  \tkzDrawPolygon[eg support](A,B,C)
  \tkzDrawSegments[eg support dashed](A,X B,Y C,Z)
  \tkzDrawSegment[eg support dashed](O,H)
  \tkzDrawCircle[eg main](N,D)

  \tkzMarkRightAngle[eg mark,size=0.18](A,X,C)
  \tkzMarkRightAngle[eg mark,size=0.18](B,Y,A)
  \tkzMarkRightAngle[eg mark,size=0.18](C,Z,B)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C)
  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](D,E,F,X,Y,Z,P,Q,R)
  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](O,H,N)

  \tkzLabelPoints[above](A)
  \tkzLabelPoints[below left](B)
  \tkzLabelPoints[below right](C)
  \tkzLabelPoints[right,yshift=-3pt](O)
  \tkzLabelPoints[above left,xshift=3pt,yshift=3pt](H)
  \tkzLabelPoints[right,yshift=-3pt](N)
  \egUseStandardPlate
\end{tikzpicture}
\end{document}