pgfplots (gnuplot): cómo trazar una función implícita para un triángulo

Question

¿Cómo puedo trazar?

\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;

junto con el triángulo equilátero

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

En este caso debería quedar algo así como un arco de círculo debajo del lado AB del triángulo.

Parece que mis métodos no son todos buenos. No soy un experto en gnuplot.

% 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}