Thank you for reporting, we will resolve it shortly
Q.
If the lengths of the sides of a triangle are in A.P. and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is :
$a < b < c$ are in A.P.
$\angle C = 2 \angle A$ (Given)
$\Rightarrow $ sinC = sin 2 A
$\Rightarrow $ sin C = 2 sin A. cos A
$\Rightarrow \; \frac{\sin C}{\sin A} = 2 \cos A$
$\Rightarrow \; \frac{c}{a} = 2 \frac{b^2 + c^2 - a^2}{2bc}$
Put $a = b - \lambda , c = b + \lambda , \lambda > 0$
$\Rightarrow \; \lambda = \frac{b}{5}$
$\Rightarrow \; a = b - \frac{b}{5} = \frac{4}{5} b, c = b + \frac{b}{5} = \frac{6b}{5}$
$\Rightarrow \; $ required ratio = $4 : 5 : 6$