A[α_,r_ ,ρ_]:=r ρ/(2 α+2(1-α)ρ) + (1-r)/(2 α ρ+2(1-α))

DD[α_,r_ ,ρ_]:=(1-r) ρ/(2 α+2(1-α)ρ) + (r)/(2 α ρ+2(1-α))

FullSimplify[A[α,r,ρ]]

FullSimplify[DD[α,r,ρ]]

Manipulate[Plot[{Star[r,ρ],r,r/(2-ρ),(1+ρ) r-.5ρ},{r,0,.5},PlotRange->{{0,0.5},{0,0.5}}],{{ρ,0.25},0,1}]

Star[r_,ρ_]:=Solve[(4ρ−2ρ ^2−2)α^3
+(2+3ρ ^2−5ρ+2r−2r ρ)α^2
+(2ρ−2r+2r  ρ−ρ ^2)α−r  ρ==0,α][[2]][[1]][[2]]

Plot3D[{A[Star[r,ρ],r,ρ],DD[Star[r,ρ],r,ρ]},{r,0,.5},{ρ,0,1},PlotLegends->{"A","D"}]