Triangles with multiple angles (5)

Let us analyze the system of equations with three unknowns (\(a\), \(b\) and \(\lambda\)) given by \[\nabla f=\lambda\nabla g\mbox{ and }g\equiv0\] that is,

\[\begin{array}{ccc} \left(\begin{array}{c} (\mathcal{S}-b)(2\mathcal{S}-2a-b)\\ (\mathcal{S}-a)(2\mathcal{S}-2b-a) \end{array}\right) & = & \lambda\left(\begin{array}{c} \frac{\delta F_{2}}{\delta a}\\ \frac{\delta F_{2}}{\delta b} \end{array}\right)\end{array}\]\[g(a,b)=0.\] Consider the two first equations \[\begin{array}{ccc} \begin{array}{r} \lambda(-2\mathcal{S}+b)\\ \lambda(2b+a) \end{array} & \begin{array}{c} =\\ = \end{array} & \begin{array}{c} (\mathcal{S}-b)(2\mathcal{S}-2a-b)\\ (\mathcal{S}-a)(2\mathcal{S}-2b-a) \end{array}\end{array}.\] Since, by triangle inequality, we have \(\mathcal{S}=\frac{a+b+c}{2}>\frac{a+a}{2}=a\) (and, similarly, \(\mathcal{S}>b\)) and the coefficients of \(\lambda\) do not vanish, we may conclude that \(2\mathcal{S}-2a-b=0\) if and only if \(2\mathcal{S}-2b-a=0.\) But the equilateral triangle (corresponding to \(2\mathcal{S}-2a-b=0=2\mathcal{S}-2b-a\)) doesn’t belong to \(\mathcal{T}_{2}\) as in this set we always have \(b>a\). Then none of the options \(2\mathcal{S}-2a-b=0\) or \(2\mathcal{S}-2b-a=0,\) which describe separately the isosceles triangles of \(\mathcal{T}_{2},\) is useful to maximize the area.

The system of three equations in the unknowns \(a,b\) and \(\lambda\) \[\begin{array}{ccc} \begin{array}{r} \lambda(-2S+b)\\ \lambda(2b+a)\\ b^{2}-2aS+ab \end{array} & \begin{array}{c} =\\ =\\ = \end{array} & \begin{array}{l} (S-b)(2S-2a-b)\\ (S-a)(2S-2b-a)\\ 0 \end{array}\end{array}\] has only one solution, where \(a\) and \(b\) are such that \[\frac{(S-b)(2S-2a-b)}{-2S+b}=\frac{(S-a)(2S-2b-a)}{2b+a}\]and \(\lambda\) equals this common value. The triangle that maximizes the area in \(\mathcal{T}_{2}\) has sides \(a\), \(b\) and \(2S-a-b\).

[1] L. Euler, Proprietates triangulorum, quorum anguli certam inter se rationem tenent, Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae Vol. XI 1765 (1767) 67-102
[2] E. Lima, Curso de Análise, IMPA, 1992
[3] I. Niven, Maxima and Minima without Calculus, MAA, Dolciani Mathematical Expositions 6, 1981