ReIm[z_]:=N[{Re[z], Im[z]}];

ReIm[ComplexInfinity]={0,1000};

Attributes[ReIm]=Listable;

LFT[mat_List][z_?NumericQ] := reim[Divide @@ (mat - {z, 1})];
Off[Power::infy, General::dbyz, Divide::infy];

polys = Table[{FaceForm[Hue[Random[], 0.6]],
    Polygon[LFT[w][triangle[]]]}, {w, G}];

Graphics[{EdgeForm[Black], polys}, PlotRange -> {{-3, 4}, {0, 2.4}},
 Frame -> True, FrameTicks -> False]