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% !TEX TS-program = lualatex
% Author Alain Matthes 2023
\documentclass{article}
\usepackage[mini]{tkz-euclide}
\usepackage{tkz-elements}
\begin{document}
The Feuerbach triangle is the triangle formed by the three points of tangency of the nine-point circle with the excircles
\begin{tkzelements}
scale = .6
z.A = point: new (0,0)
z.B = point: new (6,0)
z.C = point: new (0.8,4)
T.ABC = triangle : new ( z.A,z.B,z.C )
z.N = T.ABC.eulercenter
z.S = T.ABC.spiekercenter
T.feuerbach = T.ABC : feuerbach ()
z.Ea,z.Eb,z.Ec = get_points ( T.feuerbach )
T.excentral = T.ABC : excentral ()
z.Ja,z.Jb,z.Jc = get_points ( T.excentral )
C.JaEa = circle: new (z.Ja,z.Ea)
C.ortho = circle: radius (z.S,math.sqrt(C.JaEa: power(z.S)))
z.a = C.ortho.through
C.euler = T.ABC: euler_circle ()
C.apo = C.ortho : inversion (C.euler)
z.O = C.apo.center
z.xa,z.xb,z.xc = C.ortho : inversion (z.Ea,z.Eb,z.Ec)
\end{tkzelements}
\vspace*{2em}
\hfill
\begin{tikzpicture}
\tkzGetNodes
\tkzFillCircles[green!30](O,xa)
\tkzFillCircles[teal!30](Ja,Ea Jb,Eb Jc,Ec)
\tkzFillCircles[lightgray](S,a)
\tkzFillCircles[green!30](N,Ea)
\tkzDrawPoints(xa,xb,xc)
\tkzClipCircle(O,xa)
\tkzDrawLines[add=3 and 3](A,B A,C B,C)
\tkzDrawCircles(Ja,Ea Jb,Eb Jc,Ec S,a O,xa N,Ea)
\tkzDrawPoints(O,A,B,C,S,Ea,Eb,Ec,N)
\tkzDrawSegments[dashed](S,xa S,xb S,xc)
\tkzLabelPoints(O,N,A,B)
\tkzLabelPoints[right](S,C)
\end{tikzpicture}
\hfill
\end{document}
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