Pappus

Example 1 Pappus chain

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Wikipedia



[PDF] [TEX]

% !TEX TS-program = lualatex
% Created by Alain Matthes on 2022-01-18.
% Copyright (c) 2022 __ AlterMundus __.

\documentclass{standalone} 
\usepackage{tkz-euclide}
\usetikzlibrary{math}
\begin{document}
  \tikzmath{\xB = 6 ;
            \xC = 9 ;
            \xD = (\xC*\xC)/\xB ;
            \xJ = (\xC+\xD)/2 ; 
            \r  = \xD-\xJ; 
            \nc = 16; }

\begin{tikzpicture}[ultra thin] 
  \tkzDefPoints{0/0/A,\xB/0/B,\xC/0/C,\xD/0/D} 
  \tkzDrawCircle[diameter,fill=teal!20](A,C) 
  \tkzDrawCircle[diameter,fill=teal!30](A,B) 
  \foreach \i in {-\nc,...,0,...,\nc} {\tkzDefPoint(\xJ,2*\r*\i){J}
  \tkzDefPoint(\xJ,2*\r*\i-\r){H}
  \tkzDefCircleBy[inversion = center A through C](J,H) 
  \tkzDrawCircle[diameter,fill=teal](tkzFirstPointResult,tkzSecondPointResult)}
\end{tikzpicture}
\end{document}


Example 2 Pappus explanation



[PDF] [TEX]

\documentclass{article}
\usepackage{tkz-euclide}
\begin{document}
  
  Soit le point $D$ appartenant à la droite $(AC)$ tel que 
  \[ DB \cdot DA = AC^2\]
  alors $B$ est l'image de $D$ dans l'inversion de centre $A$ et puissance $AC^2$.
  Les demi-cercles de diamètre $[AB]$ et$[AC]$ passent par le pôle $A$. Ils ont pour images les demi-droites $\mathcal{L'}$ et $\mathcal{L}$.
  
Les cercles de centre $J_i$ et de diamètre $S_iT_i$ ont pour images les cercles de diamètre $S'_iT'_i$.

  \pgfmathsetmacro{\xB}{6}%
  \pgfmathsetmacro{\xC}{9}%
  \pgfmathsetmacro{\xD}{(\xC*\xC)/\xB}%
  \pgfmathsetmacro{\xJ}{(\xC+\xD)/2}%
  \pgfmathsetmacro{\r}{\xD-\xJ}%
  \pgfmathsetmacro{\nc}{2}%
  
\begin{tikzpicture}[scale=1,ultra thin]
  \tkzDefPoints{0/0/A,\xB/0/B,\xC/0/C,\xD/0/D}
  \tkzDefPointBy[rotation = center C angle -90](B)  \tkzGetPoint{c}
  \tkzDefPointBy[rotation = center A angle 90](C)  \tkzGetPoint{a}
  \tkzDefPointBy[rotation = center D angle -90](C)  \tkzGetPoint{d}
  \tkzDrawLines[add=0 and 2.25](C,c)
  \tkzDrawLines[add=0 and 1.5](D,d)
  \tkzDrawSemiCircle[diameter](A,C)
  \tkzDrawSemiCircle[diameter](A,B)
  \tkzDrawSemiCircle[diameter](B,C)
  \tkzDrawSemiCircle[diameter](C,D)
  \tkzDrawArc[red](A,C)(a)
  \tkzDrawPoints(A,B,C,D)
  \tkzLabelPoints(A,B,C,D)
  \tkzLabelLine[left,pos=3](C,c){$\mathcal{L}$}
  \tkzLabelLine[right,pos=2.5](D,d){$\mathcal{L'}$}
  \foreach \i in {1,...,\nc}
{
  \tkzDefPoint(\xJ,2*\r*\i){J} 
  \tkzDefPoint(\xJ,2*\r*\i-\r){H}
  \tkzDefCircleBy[inversion = center A through C](J,H)\tkzGetPoints{J'}{H'}
  \tkzInterLC(A,J)(J,H) \tkzGetPoints{S}{T}
  \tkzDefPointsBy[inversion = center A through C](H,S,T){H',S',T'}
  \tkzDrawCircle(J,H)
  \tkzDrawCircle[diameter](S',T')
  \tkzDrawLines[dashed,add = 0 and .15](A,T A,S A,H) 
  \tkzDrawPoints(J,H,H',S,S',T,T') 
  \tkzLabelPoint(J){$J_\i$}
  \tkzLabelPoint(S){$S_\i$}
  \tkzLabelPoint(T){$T_\i$}
  \tkzLabelPoint(H){$H_\i$}
  \tkzLabelPoint(S'){$S'_\i$}
  \tkzLabelPoint(T'){$T'_\i$}
  \tkzLabelPoint(H'){$H'_\i$}
  }
\end{tikzpicture}
\end{document}