In $\triangle ABC$,if $r_1 = 2r_2 = 3r_3$,then $b : c$ equals

Let $ABC$ be a triangle such that $\angle ACB = \frac{\pi}{6}$ and let $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively. The value$(s)$ of $x$ for which $a = x^2+x+1, b = x^2-1$ and $c = 2x+1$ is (are)

In a $\triangle ABC$,the expression $\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4b^2c^2}$ equals:

In a $\triangle ABC$,if $a+3b=3c$,then $\sin \frac{A}{2} =$

In a $\Delta ABC$,$a, c, A$ are given and $b_1, b_2$ are two values of the third side $b$ such that $b_2 = 2b_1$. Then $\sin A = $

If in triangle $ABC$,with usual notations $\sin \frac{A}{2} \cdot \sin \frac{C}{2} = \sin \frac{B}{2}$ and $2s$ is the perimeter of the triangle,then the value of $s$ is