Mathematica code:
coords =1/2{{1,-(Sqrt[3]/3)},{0,(2Sqrt[3])/3},{-1,-(Sqrt[3]/3)}};
colors ={Red,Green,Blue};
Fold[ImageAdd,First@#,Rest@#]&@
MapThread[Graphics[{##},PlotRange->1.8,Background->Black]&,
{colors,Disk/@ coords}]
Clear[coord, colors]
Manipulate[
With[{c1 ={x1, y1}, c2 ={x2, y2}, c3 ={x3, y3}},
Module[{pts1, pts2, pts3},
{pts1, pts2, pts3}=
Quiet@Solve[{Tr[({x, y}-#1)^2]==#2^2,
Tr[({x, y}-#3)^2]==#4^2},{x, y},Reals]&@@@{{c1, r1,
c2, r2},{c1, r1, c3, r3},{c2, r2, c3, r3}};
Fold[ImageAdd,First@#,Rest@#]&@
MapThread[
Graphics[{##,{PointSize[Large],
Point[If[#=={},{},{x, y}/.#//Re]]&/@{pts1, pts2,
pts3}}}, Axes->1,AxesStyle->White,PlotRange->6,
Background->GrayLevel[0.02]]&,{{Red,Green,Blue},
Disk@@@{{c1, r1},{c2, r2},{c3, r3}}}]]],
{{x1,1},-4, 4},{{y1,-1/Sqrt[3]},-4,4},{{r1,2},0,4},
{{x2,0},-4,4},{{y2,2/Sqrt[3]},-4,4},{{r2,2},0,4},
{{x3,-1},-4, 4},{{y3,-1/Sqrt[3]},-4,4},{{r3,2},0,4}]