Pensieve header: A constrained minimation for the hard part of the isoperimetric inequallity.

f=(1/2)a b Sin[α];
c=a+b-(Sqrt[(a^2)+(b^2)-2a b Cos[α]]);
g=f-λ c;
grad={(∂_a)g, (∂_b)g}

sols=Solve[grad=={0,0}, {a,b}]

a==b /. sols