Apollonius circle
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\documentclass{standalone}
\usepackage{tkz-euclide}
\usepackage{tkz-elements}
\begin{document}
\begin{elements}
scale = 0.5
z.A = point: new (0,0)
z.B = point: new (6,0)
z.C = point: new (0.8,4) ; id ()
T = triangle: new(A,B,C)
z.O = T: circum_center ()
_,_,z.Mc = T: medial_tr ()
z.Ja,z.Xa,_,_,
z.Jb,z.Xb,_,_,
z.Jc,z.Xc,_,_ = T: circle_ex_all()
z.N = T: euler_center ()
z.L = T: lemoine_center ()
z.S = T: spieker_center () ; id ()
z.Ap = intersection_ll_ (O,L,N,S); id ()
z.x = ll_from_ (Ap,N,Mc) ; id ()
z.rAp = intersection_ll_ (Ap,x,Mc,S)
\end{elements}
\begin{tikzpicture}
\tkzGetNodes
\tkzDrawLines[red](Ap,L N,Ap Ap,rAp Mc,rAp Mc,N)
\tkzDrawLines[teal,add=1 and 1](A,B A,C B,C)
\tkzDrawCircles(Ja,Xa Jb,Xb Jc,Xc)
\tkzDrawCircle[purple](Ap,rAp)
\tkzDrawPoints(A,B,C,Ja,Jb,Jc)
\tkzDrawPoints[red](N,O,L,S,Ap,rAp)
\end{tikzpicture}
\end{document}