RandomInteger[{1,3}]

(Subscript[A, 1]) = {-1,0}; (Subscript[A, 2])= {0,1}; (Subscript[A, 3]) = {1,0};

Pt = {0.,0.};
Pts = {Pt}

For[j=1, j<10000, j++, 
v=RandomInteger[{1,3}];
CornerPt = (Subscript[A, v]);
Pt = Simplify[(1/2)(Pt+CornerPt)];
Pts = Join[Pts,{Pt},1];
]

ListPlot[Pts,PlotStyle->{{Red}}, AspectRatio->1]

ListPlot[Pts,PlotStyle->{{Red}}, PlotRange->{{0,1},{0,1/2}}, AspectRatio->1]

(Subscript[B, 1]) = {-1,-1}; (Subscript[B, 2])= {1,1}; (Subscript[B, 3]) = {-1,1}; (Subscript[B, 4]) = {1,-1};

Pt = {0.,0.};
Pts = {Pt}

For[j=1, j<30000, j++, 
v=RandomInteger[{1,4}];
CornerPt = (Subscript[B, v]);
Pt = Simplify[(1/3)(Pt+CornerPt)];
Pts = Join[Pts,{Pt},1];
]

ListPlot[Pts,PlotStyle->{{Red}}, AspectRatio->1]

ListPlot[Pts,PlotStyle->{{Red}}, AspectRatio->1, PlotRange->{{0,0.5},{0,0.5}}]

ListPlot[Pts,PlotStyle->{{Red}}, AspectRatio->1, PlotRange->{{0.1,0.3},{0.1,0.3}}]

(Subscript[A, 1]) = {-1,0}; (Subscript[A, 2])= {0,1}; (Subscript[A, 3]) = {1,0};

Pt = {0.,0.};
Pts = {Pt};
α = (97/200);

For[j=1, j<10000, j++, 
v=RandomInteger[{1,3}];
CornerPt = (Subscript[A, v]);
Pt = Simplify[(1-α) Pt + α CornerPt];
Pts = Join[Pts,{Pt},1];
]

ListPlot[Pts,PlotStyle->{{Red}}, AspectRatio->1]

PlotTriangle[α_] := Block[{},Pt = {0.,0.};
Pts = {Pt};
For[j=1, j<10000, j++, 
v=RandomInteger[{1,3}];
CornerPt = (Subscript[A, v]);
Pt = Simplify[(1-α) Pt + α CornerPt];
Pts = Join[Pts,{Pt},1];
];
Return[Pts]];

Manipulate[ListPlot[PlotTriangle[α],PlotStyle->{{Red}}, AspectRatio->1,PlotRange->{{-1,1},{0,1}}, Axes->False, ImageSize->{400,400}],{{α,0.3},0.3,0.4,0.01},SaveDefinitions->True]