If in a $\Delta ABC$,$b = 2$,$c = \sqrt{3}$,$\angle A = \frac{\pi}{6}$,then $R = $?

In $\triangle ABC$,if $a^2-c^2=b(b-c)$,$\sqrt{2}a=2b-c$ and $R=\frac{1}{\sqrt{3}}$,then $b=$

In a triangle $ABC$,if $r_1=2 r_2=3 r_3$,then $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=$

The sides of a triangle are in the ratio $1 : \sqrt{3} : 2$. Then the angles are in the ratio

In a $\triangle ABC$,if $3a = b + c$,then $\cot \frac{B}{2} \cot \frac{C}{2}$ is equal to :

Points $D$ and $E$ are taken on the side $BC$ of a triangle $ABC$ such that $BD = DE = EC$. If $\angle BAD = x$,$\angle DAE = y$,and $\angle EAC = z$,then the value of $\frac{\sin(x + y)\sin(y + z)}{\sin x \sin z}$ is: