In $\triangle ABC$,with usual notations,if $\frac{1}{b+c}+\frac{1}{c+a}=\frac{3}{a+b+c}$,then $m \angle C$ is equal to

If the angles $A, B$ and $C$ of a triangle are in $A.P.$ and if $a, b$ and $c$ denote the length of the sides opposite to $A, B$ and $C$ respectively,then the value of $\frac{a}{b} \sin 2B + \frac{b}{a} \sin 2A$ is

The lengths of the sides of a triangle are $13, 14, \text{ and } 15$. If $R$ and $r$ respectively denote the circumradius and inradius of this triangle,then $8R - r = $

In a triangle $ABC$,let $AB = \sqrt{23}$,$BC = 3$,and $CA = 4$. Then the value of $\frac{\cot A + \cot C}{\cot B}$ is:

Let $ABC$ be an acute-angled triangle and let $D$ be the mid-point of $BC$. If $AB = AD$,then $\tan(B) / \tan(C)$ equals

In $\triangle ABC$,if the line joining the circumcentre $(O)$ and the incentre $(I)$ is parallel to $BC$,then $\cos B + \cos C = $