pgfplots (gnuplot): So zeichnen Sie eine implizite Funktion für ein Dreieck

Question

Wie kann ich plotten

\pgfmathsetmacro\xA{0}
\pgfmathsetmacro\yA{0}
\pgfmathsetmacro\xB{1}
\pgfmathsetmacro\yB{0}
\pgfmathsetmacro\xC{0.5}
\pgfmathsetmacro\yC{0.5*sqrt(3)}

f(x,y)=4*(  (x -\xA)^2 + (y - \yA)^2)*((x -\xB)^2 + (y -\yB)^2  ) 
-(   ((x -\xC)^2 + (y -\yC)^2) 
    -((x -\xA)^2 + (y -\yA)^2) 
    -((x -\xB)^2 + (y -\yB)^2)     )^2;

zusammen mit dem gleichseitigen Dreieck

\addplot[no marks] coordinates {(\xA,\yA)  (\xB,\yB) (\xC,\yC) (\xA,\yA) };

In diesem Fall sollte so etwas wie ein Kreisbogen unterhalb der Dreiecksseite AB entstehen.

Es scheint, dass meine Methoden alle nicht gut sind. Ich bin kein Gnuplot-Experte.

% arara: pdflatex: {shell: yes}
\documentclass[margin=5mm, tikz]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}

% Ecken
\pgfmathsetmacro\xA{0}
\pgfmathsetmacro\yA{0}
\pgfmathsetmacro\xB{1}
\pgfmathsetmacro\yB{0}
\pgfmathsetmacro\xC{0.5}
\pgfmathsetmacro\yC{0.5*sqrt(3)}

% A(0,0); B(1,0); C(0.5, 0.5*sqrt(3))

\begin{tikzpicture}
\begin{axis}[axis equal, 
title={$|AX| + |BX| = |CX|$}, 
]
\addplot +[%smooth,
no markers,
raw gnuplot,
thick,
%empty line = jump % not strictly necessary,
] gnuplot {
f(x,y)=4*((x -\xA)^2 + (y - \yA)^2)*((x -\xB)^2 + (y -\yB)^2) 
-( ((x -\xC)^2 + (y -\yC)^2) 
-((x -\xA)^2 + (y -\yA)^2) 
-((x -\xB)^2 + (y -\yB)^2) )^2;
set cntrparam levels discrete 0,0;
set isosample 100,100;
set size square;
set view equal xy;
set cont base;
unset surface;
splot f(x,y);
};
\addlegendentry{$X$}
\addplot[no marks] coordinates {(\xA,\yA)  (\xB,\yB) (\xC,\yC) (\xA,\yA) };

\draw[fill=white] (\xA,\yA) circle (1.75pt) node[anchor=north east]{$A(\xA,\yA)$};
\draw[fill=white] (\xB,\yB) circle (1.75pt) node[anchor=north]{$B(\xB,\yB)$};
\draw[fill=white] (\xC,\yC) circle (1.75pt) node[anchor=east]{$C(\xC,\yC)$};
\end{axis}
\end{tikzpicture}


\begin{tikzpicture}
  \begin{axis}[
view = {0}{90},
axis equal,
%     restrict x to domain = -2.2:3 
]
\addplot3 [contour gnuplot = {labels = false}, samples=10]
{ 
2*sqrt((x -\xA)^2 + (y - \yA)^2)*sqrt((x -\xB)^2 + (y -\yB)^2) 
-( ((x -\xC)^2 + (y -\yC)^2) 
-((x -\xA)^2 + (y -\yA)^2) 
-((x -\xB)^2 + (y -\yB)^2) )^1
};
\addlegendentry{$X$}

\addplot[no marks] coordinates {(\xA,\yA)  (\xB,\yB) (\xC,\yC) (\xA,\yA) };

\draw[fill=white] (\xA,\yA) circle (1.75pt) node[anchor=north east]{$A(\xA,\yA)$};
\draw[fill=white] (\xB,\yB) circle (1.75pt) node[anchor=north]{$B(\xB,\yB)$};
\draw[fill=white] (\xC,\yC) circle (1.75pt) node[anchor=east]{$C(\xC,\yC)$};
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
  \begin{axis}[
      axis equal,
%      restrict y to domain = -4:4,
    ]
    \addplot + [
      no markers,
      raw gnuplot,
      thick,
      ]
      gnuplot {
f(x,y)=4*((x -\xA)^2 + (y - \yA)^2)*((x -\xB)^2 + (y -\yB)^2) 
-( ((x -\xC)^2 + (y -\yC)^2) 
-((x -\xA)^2 + (y -\yA)^2) 
-((x -\xB)^2 + (y -\yB)^2) )^2;
        set contour base;
        set cntrparam levels discrete 0.01;
        unset surface;
        set view map;
        set isosamples 300;
        splot f(x,y);
      };
\addlegendentry{$X$}

\addplot[no marks] coordinates {(\xA,\yA)  (\xB,\yB) (\xC,\yC) (\xA,\yA) };

\draw[fill=white] (\xA,\yA) circle (1.75pt) node[anchor=north east]{$A(\xA,\yA)$};
\draw[fill=white] (\xB,\yB) circle (1.75pt) node[anchor=north]{$B(\xB,\yB)$};
\draw[fill=white] (\xC,\yC) circle (1.75pt) node[anchor=east]{$C(\xC,\yC)$};
\end{axis}
\end{tikzpicture}


\end{document}