If in $\triangle ABC$ $a=\dfrac{\overline{BC}}{2},\dfrac{\overline{AB}}{\overline{AC}}=\dfrac{c}{b}$ then the maximum area of the triangle $= \dfrac{2a^2bc}{|b^2-c^2|}$

Proof:

$\overline{BC}=2a$ $\dfrac{\overline{AB}}{\overline{AC}}=\dfrac{c}{b}$ let $\overline{AB}=2cx,\overline{AC}=2bx$ $\Rightarrow s=a+x(b+c)$ $\Rightarrow s-\overline{BC}=x(b+c)-a$ $\Rightarrow s-\overline{AB}=a+x(b-c)$ $\Rightarrow s-\overline{AC}=a-x(b-c)$ If the area …