Pappus and Radical Axis for Problem 5021

Completed figures

Two directly similar triangles OAB and OCD. Lines AE and DF meet at P, while AC and BD meet at Q. The points O, Q, and P are collinear. The two circumcircles have centers K and L, and OP is perpendicular to KL. — The Pappus configuration and two circumcircles used in the generalized solution.
Complete source

Pappus and radical-axis configuration

% Figure for Crux Mathematicorum Problem 5021
% Site: Elementary Geometry
% Website: https://elementarygeometry.org/
% Drawing tool: tkz-euclide
% Output format: SVG
% Last verified: 2026-08-30
%
% Editable source for public/diagrams/crux-5021-pappus-radical-axis.svg.
% This sanitized drawing contains no source problem text or contact information.
% Accent target: radical axis OP
% Publication Profile: scale 1.0.
% The web build does not compile TeX; export and review the SVG separately.
\documentclass[tikz,border=8pt]{standalone}
\usepackage{tkz-euclide}
\usetikzlibrary{intersections,calc,angles,quotes}

\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(1.6,3.2){O}
  \tkzDefPoint(0,0){A}
  \tkzDefPoint(4.8,0){B}

  \tkzDefPointBy[rotation=center O angle 20](A)\tkzGetPoint{CC}
  \tkzDefMidPoint(O,CC)\tkzGetPoint{C}

  \tkzDefPointBy[rotation=center O angle 20](B)\tkzGetPoint{DD}
  \tkzDefMidPoint(O,DD)\tkzGetPoint{D}

  \tkzInterLL(C,D)(O,B)\tkzGetPoint{E}
  \tkzInterLL(A,B)(O,C)\tkzGetPoint{F}
  \tkzInterLL(F,D)(A,E)\tkzGetPoint{P}
  \tkzInterLL(A,C)(B,D)\tkzGetPoint{Q}

  \tkzDefTriangleCenter[circum](O,A,B)\tkzGetPoint{K}
  \tkzDefTriangleCenter[circum](O,C,D)\tkzGetPoint{L}

  \tkzDrawSegments[eg support](O,A A,B B,O O,C C,D D,O A,C)
  \tkzDrawSegments[eg support dashed](F,P A,P C,F C,Q B,Q)

  \tkzDefCircle[circum](O,A,B)\tkzGetPoints{cOAB}{r1}
  \tkzDrawCircle[eg main](cOAB,r1)
  \tkzDefCircle[circum](O,C,D)\tkzGetPoints{cOCD}{r2}
  \tkzDrawCircle[eg main](cOCD,r2)

  \tkzDrawLine[eg accent,add=0.35 and 0.1](O,P)
  \tkzDrawLine[eg support dashed,add=1.25 and 0.75](K,L)

  \tkzDrawPoints[color=egNavy,fill=egNavy,size=1.6](O,A,B,C,D,E,F,P,Q,K,L)

  \tkzLabelPoints[above](O)
  \tkzLabelPoints[left](A,C)
  \tkzLabelPoints[right](B,D)
  \tkzLabelPoints[below](E,F)
  \tkzLabelPoints[above](P,Q)
  \tkzLabelPoints[below left](K,L)
  \egUseStandardPlate
\end{tikzpicture}
\end{document}
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Read Problem 5021 and the proof