p1={1,3}

p2={2,5}

t*p1

(1-t)*p2

t*p1+(1-t)*p2

Expand[t*p1+(1-t)*p2]

{p1,p2}

ParametricPlot[t*p1+(1-t)*p2,{t,0,1},Epilog->Point[{p1,p2}]]

ParametricPlot[t*p1+(1-t)*p2,{t,-1,2},Epilog->Point[{p1,p2}]]

Graphics[{Red,Thick,Line[{p1,p2}]}]

Show[{ParametricPlot[t*p1+(1-t)*p2,{t,-1,2},Epilog->Point[{p1,p2}]],Graphics[{Red,Thick,Line[{p1,p2}]}]}]

Table[{Cos[k*2Pi/8],Sin[k*2Pi/8]},{k,1,8}]

Graphics[{Circle[{0,0},1],Line[Table[{Cos[k*2Pi/8],Sin[k*2Pi/8]},{k,0,8}]]},ImageSize->200,Axes->True]

Graphics[{Circle[{0,0},1],Line[Table[{Cos[2Pi/16+k*2Pi/8],Sin[2Pi/16+k*2Pi/8]},{k,0,8}]]},ImageSize->200,Axes->True]

Graphics[Line[{{1,3},{2,5},{3,4},{1,3}}]]

valerio@mat.uniroma3.it

lista=Table[{Cos[k*2Pi/10],Sin[k*2Pi/10]},{k,0,10}]

N[lista]

Map[#+{2,0}&,N[lista]]

(* lista di coordinate dei punti vertici di un poligono regolare di 10 lati *)

lista=Table[{Cos[k*2Pi/10],Sin[k*2Pi/10]},{k,0,10}]

Graphics[{Circle[{0,0},1],Line[lista],Red,Point[lista]},ImageSize->150]

Graphics[{Circle[{0,0},1],Line[lista],Red,Point[lista]},ImageSize->150]

Show[Table[Graphics[{Circle[{0,0}+k*{2,0},1],Line[Map[#+k*{2,0}&,lista]],Red,Point[Map[#+k*{2,0}&,lista]]}],{k,0,3}],ImageSize->450,Axes->True,PlotRange->{{-2,8},{-2,2}}]

Show[Table[Graphics[{Circle[{0,0}+k*{2,0},1],Line[Map[#+k*{2,0}&,lista]],Red,Point[Map[#+k*{2,0}&,lista]]}],{k,0,3}],ImageSize->450,Axes->True,PlotRange->{{-2,8},{-2,2}}]

(* matrice di rotazione intorno all'origine di un angolo a in senso antiorario *)

rt[a_]:={{Cos[a],-Sin[a]},{Sin[a],Cos[a]}}

MatrixForm[rt[a]]

Show[Table[Graphics[{Circle[{0,0}+k*{2,0},1],Line[Map[#+k*{2,0}&,lista]],Red,Point[Map[rt[Pi/10].#+k*{2,0}&,lista]]}],{k,0,3}],ImageSize->450,Axes->True,PlotRange->{{-2,8},{-2,2}}]

Show[Table[Graphics[{Line[Map[rt[Pi/10].#+k*{2,0}&,lista]],Red,Point[Map[rt[Pi/10].#+k*{2,0}&,lista]]}],{k,0,3}],ImageSize->450,Axes->True,PlotRange->{{-2,8},{-2,2}}]

N[lista]

N[lista[[2]]]

lista=Table[{Cos[k*2Pi/10],Sin[k*2Pi/10]},{k,0,10}]

N[lista[[1]]+lista[[2]]]/2

(* distanza tra due punti nel piano *)

dist[{x1_,y1_},{x2_,y2_}]:=Sqrt[(x1-x2)^2+(y1-y2)^2]

?Circle

Module[{lista=Table[{Cos[k*2Pi/10],Sin[k*2Pi/10]},{k,0,10}],med,rad},
med=N[lista[[1]]+lista[[2]]]/2;
rad=1/2Norm[lista[[1]]-lista[[2]]];
ParametricPlot[med+rad{Cos[t],Sin[t]},{t,0,2Pi},Epilog->{Circle[med,dist[{1,0},med],{-2/5Pi,-2/5Pi+Pi}],Line[lista],Red,Point[lista],Blue,Point[med]},ImageSize->250,PlotRange->{{-2,2},{-2,2}}]]

Module[{lista=Table[{Cos[(2k-1)Pi/10],Sin[(2k-1)Pi/10]},{k,0,10}],med,rad},
med=N[lista[[1]]+lista[[2]]]/2;
rad=1/2Norm[lista[[1]]-lista[[2]]];
ParametricPlot[med+rad{Cos[t],Sin[t]},{t,-Pi/2,Pi/2},Epilog->{Line[lista],Red,Point[lista],Blue,Point[med]},ImageSize->250,PlotRange->{{-2,2},{-2,2}}]]

Module[{lista,med,rad,n=6},
lista=Table[{Cos[(2k-1)Pi/n],Sin[(2k-1)Pi/n]},{k,0,n}];
med=N[lista[[1]]+lista[[2]]]/2;
rad=1/2Norm[lista[[1]]-lista[[2]]];
ParametricPlot[Table[rt[2k*Pi/n].(med+rad{Cos[t+Mod[k,2]*Pi],Sin[t+Mod[k,2]*Pi]}),{k,0,n-1}],{t,-Pi/2,Pi/2},Epilog->{Line[lista],Red,Point[lista],Blue,Point[med]},ImageSize->250,PlotRange->{{-2,2},{-2,2}}]]

Module[{med=N[lista[[1]]+lista[[2]]]/2},Graphics[{Circle[med,dist[{1,0},med],{-2/5Pi,-2/5Pi+Pi}],Line[{{0,0},lista[[2]]}],Line[lista],Red,Point[lista],Blue,Point[med]},ImageSize->250,Axes->True,PlotRange->{{-2,2},{-2,2}}]]

Module[{med=N[lista[[1]]+lista[[2]]]/2},Graphics[{
Table[Circle[rt[k*2Pi/10].med,dist[{1,0},med],{-2/5Pi+k*2Pi/10,-2/5Pi+k*2Pi/10+Pi}],{k,1,10}]
,Line[{{0,0},lista[[2]]}],Line[lista],Red,Point[lista],Blue,Point[med]},ImageSize->250,Axes->True,PlotRange->{{-2,2},{-2,2}}]]

Module[{med=N[lista[[1]]+lista[[2]]]/2},Graphics[{
Table[Circle[rt[k*2Pi/10].med,dist[{1,0},med],{-2/5Pi+k*2Pi/10+Pi,-2/5Pi+k*2Pi/10+Pi+Pi}],{k,1,10}]
,Line[{{0,0},lista[[2]]}],Line[lista],Red,Point[lista],Blue,Point[med]},ImageSize->250,Axes->True,PlotRange->{{-2,2},{-2,2}}]]

Manipulate[
lista=Table[{Cos[k*2Pi/n],Sin[k*2Pi/n]},{k,0,n}];Module[{med=N[lista[[1]]+lista[[2]]]/2},Graphics[{
Table[Circle[rt[k*2Pi/n].med,dist[{1,0},med],{-(n-2)/(2n)*Pi+k*2Pi/n+Pi,-(n-2)/(2n)*Pi+k*2Pi/n+Pi+Pi}],{k,1,n,2}],Table[Circle[rt[k*2Pi/n].med,dist[{1,0},med],{-(n-2)/(2n)*Pi+k*2Pi/n,-(n-2)/(2n)*Pi+k*2Pi/n+Pi}],{k,0,n,2}],Red,Point[lista]},ImageSize->250,Axes->True,PlotRange->{{-2,2},{-2,2}}]],{n,4,40,2}]