## Triangles with multiple angles

### 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