Tangents, Angle Bisectors, and a Common Point

% Figure for a Classical Theorem Note
% Article: Tangents, Angle Bisectors, and a Common Point
% Site: Elementary Geometry
% Website: https://elementarygeometry.org/
% Drawing tool: tkz-euclide
% Output format: SVG
% Standard plate: 5:4
% Last verified: 2026-09-06
% Accent target: none
%
% Editable source for public/diagrams/tangents-angle-bisectors-common-point.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]
  % A uniform similarity of the canonical ABC used by the site's triangle figures.
  \tkzDefPoint(0,0){S}
  \tkzDefPoint(-0.5,4.2){A0}
  \tkzDefPoint(-3,0){B0}
  \tkzDefPoint(3,0){C0}
  \tkzDefPointBy[homothety=center S ratio 0.4](A0)\tkzGetPoint{A}
  \tkzDefPointBy[homothety=center S ratio 0.4](B0)\tkzGetPoint{B}
  \tkzDefPointBy[homothety=center S ratio 0.4](C0)\tkzGetPoint{C}

  % Circumcircle and its three tangent intersections.
  \tkzDefCircle[circum](A,B,C)\tkzGetPoint{O}
  \tkzDefLine[orthogonal=through A](O,A)\tkzGetPoint{tA}
  \tkzDefLine[orthogonal=through B](O,B)\tkzGetPoint{tB}
  \tkzDefLine[orthogonal=through C](O,C)\tkzGetPoint{tC}
  \tkzInterLL(B,tB)(C,tC)\tkzGetPoint{P}
  \tkzInterLL(C,tC)(A,tA)\tkzGetPoint{Q}
  \tkzInterLL(A,tA)(B,tB)\tkzGetPoint{R}

  % Internal angle-bisector traces.
  \tkzDefTriangleCenter[in](A,B,C)\tkzGetPoint{I}
  \tkzInterLL(A,I)(B,C)\tkzGetPoint{D}
  \tkzInterLL(B,I)(C,A)\tkzGetPoint{E}
  \tkzInterLL(C,I)(A,B)\tkzGetPoint{F}

  % Construct the concurrence point from two lines; the third passes through it.
  \tkzInterLL(P,D)(Q,E)\tkzGetPoint{T}

  \tkzDrawCircle[eg support](O,A)
  \tkzDrawPolygon[eg support](P,Q,R)
  \tkzDrawSegments[eg support dashed](A,D B,E C,F)
  \tkzDrawSegments[eg main](A,B B,C C,A)
  \tkzDrawSegments[eg support dashed,draw=egNavy](P,T Q,T)
  % Draw the third line from the independently constructed points R and F;
  % its extension passes through T=PD\cap QE.
  \tkzDrawLine[eg support dashed,draw=egNavy,add=0 and 0.62](R,F)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](A,B,C,D,E,F,P,Q,R,T)
  \tkzLabelPoints[above](A)
  \tkzLabelPoints[below left](B)
  \tkzLabelPoints[below right](C)
  \tkzLabelPoints[below right,yshift=-2pt](D)
  \tkzLabelPoints[right](E)
  \tkzLabelPoints[left](F)
  \tkzLabelPoints[left](P)
  \tkzLabelPoints[right](Q)
  \tkzLabelPoints[left](R)
  \tkzLabelPoints[below left,xshift=-1pt,yshift=-1pt](T)
  \egUseStandardPlate
\end{tikzpicture}
\end{document}