FullForm[{a,b,c}]

FullForm[b*c]

Apply[Plus,b*c]

FullForm[a+b*c]

FullForm[(a+b)*c]

Apply[Plus,(a+b)*c]

perimetro[A_,B_,C_]:=Module[{a=dist[B,C],b=dist[A,C],c=dist[A,B]},Apply[Plus,{a,b,c}]]

incentro[A_,B_,C_]:=Module[{a=dist[B,C],b=dist[A,C],c=dist[A,B]},1/perimetro[A,B,C]*{Map[#[[1]]&,{A,B,C}].{a,b,c},Map[#[[2]]&,{A,B,C}].{a,b,c}}]

Module[{A,B,C,inc},
Manipulate[inc=incentro[A,B,C];Graphics[{Circle[inc,dist[inc,piede[A,C,inc]]],Line[{A,inc}],Line[{B,inc}],Line[{C,inc}],
Red,PointSize[.02],Point[{A,B,C}],
Blue,Opacity[0.05],EdgeForm[Thick],Polygon[{A,B,C}]},PlotRange->{{-3,3},{-3,3}},ImageSize->250],
{{A,{1,2}},{-3,-3},{3,3},Locator},{{B,{3,1}},{-3,-3},{3,3},Locator},{{C,{-1,0}},{-3,-3},{3,3},Locator}]
]

incentro[A_,B_,C_]:=Module[{a=dist[B,C],b=dist[A,C],c=dist[A,B]},1/perimetro[A,B,C]*{Map[#[[1]]&,{A,B,C}].{a,b,c},Map[#[[2]]&,{A,B,C}].{a,b,c}}]

excentri[A_,B_,C_]:=Module[{a=dist[B,C],b=dist[A,C],c=dist[A,B],vec},
vec={{a,b,-c},{a,-b,c},{-a,b,c}};Table[1/Apply[Plus,vec]{Map[#[[1]]&,{A,B,C}].vec[[k]],Map[#[[2]]&,{A,B,C}].vec[[k]]},{k,1,3}]]

perimex[P1_,P2_,P3_,k_]:=Module[{a=Norm[P2-P3],b=Norm[P1-P3],c=Norm[P1-P2],perm=Permutations[{1,1,-1}]},Apply[Plus,DiagonalMatrix[perm[[k]]].{a,b,c}]]

area[P1_,P2_,P3_]:=Module[{a=Norm[P2-P3],b=Norm[P1-P3],c=Norm[P1-P2],s=1/2perimetro[P1,P2,P3]},Simplify[Sqrt[s(s-a)(s-b)(s-c)]]]

eqretta[{x1_,y1_},{x2_,y2_}]:=If[x1==x2,x==x1,y==Expand[y1+(y2-y1)(x-x1)/(x2-x1)]];

eqrettabis[P1_,P2_,P3_]:=
eqretta[P1,incentro[P1,P2,P3]]

piedebis[P1_,P2_,P3_]:=With[{r=eqretta[P2,P3]},{x,y}/.Solve[{r,eqrettabis[P1,P2,P3]},{x,y}][[1]]]

Manipulate[Module[{exc=excentri[P1,P2,P3]},Show[{Graphics[{{LightGreen,Polygon[exc]},{LightRed,Polygon[{P1,P2,P3}]},Line[{P1,piedebis[P1,P2,P3]}],Line[{P2,piedebis[P2,P3,P1]}],Line[{P3,piedebis[P3,P2,P1]}],{Red,PointSize[0.02],Point[{P1,P2,P3}]},{Blue,PointSize[0.02],Point[incentro[P1,P2,P3]],Circle[incentro[P1,P2,P3],2area[P1,P2,P3]/perimetro[P1,P2,P3]]},{Blue,PointSize[0.02],Point[exc],Table[Circle[exc[[k]],2area[P1,P2,P3]/perimex[P1,P2,P3,k]],{k,1,3}]},Text[Style["(Subscript[P, 1])",Medium],P1,{-2,-2}],Text[Style["(Subscript[P, 2])",Medium],P2,{-2,-2}],Text[Style["(Subscript[P, 3])",Medium],P3,{-2,-2}]},Axes->True,PlotRange->{{-5,5},{-5,5}},ImageSize->400],ParametricPlot[{t*P1+(1-t)*P2,t*P1+(1-t)*P3,t*P3+(1-t)*P2},{t,-5,5}],ParametricPlot[{t*exc[[1]]+(1-t)*exc[[2]],t*exc[[1]]+(1-t)*exc[[3]],t*exc[[3]]+(1-t)*exc[[2]]},{t,-5,5}]}]],{{P1,{2,-2}},Locator,Appearance->None},{{P2,{0,1}},Locator,Appearance->None},{{P3,{-1,-1}},Locator,Appearance->None}]