Circumcircles

Clear[circle];
circle[p1_, p2_, p3_] := circle[p1, p2, p3] = Module[{soln},
    soln =
     Quiet@Solve[
       Join[Map[(#[[1]] - a)^2 + (#[[2]] - b)^2 == r^2 &, {p1, p2,
          p3}], {r > 0}], {a, b, r}];
    ({a, b, r} /. soln[[1]])
    ];
n = 30;
ps = RandomReal[{-1, 1}, {n, 2}];
frame[m_] :=
 Module[{triples},
  triples = Subsets[ps[[;; m]], {3}];
  Graphics[{
    Table[
     Block[{a, b, r},
      {a, b, r} = circle @@ triples[[i]];
      {Opacity[0.01 + 0.1 Tanh[0.5 r] + 2/m],
       ColorData["DarkRainbow"][Tanh[0.5 r]], Circle[{a, b}, r]}
      ]
     , {i, Length@triples}],
    {PointSize[Large], White, Point[ps[[;; m]]], Darker@Purple,
     PointSize[Medium], Point[ps[[;; m]]], Red, PointSize[Large],
     Point[ps[[m]]]}
    }, PlotRange -> 2]
  ]; frame[n];
Manipulate[
frame[m],
{m,1,n,1}
]