In a triangle $ABC$ with usual notations,if $\cot \frac{A}{2} = \frac{b+c}{a}$,then the triangle $ABC$ is

In a $\Delta ABC$,if $C = 90^{\circ}$,then $\frac{a^{2} + b^{2}}{a^{2} - b^{2}} \sin(A - B)$ is equal to:

In a triangle,the lengths of the sides are integers. Suppose that the length of one side is $1$,and the longest altitude is twice the shortest altitude. Let $R$ and $r$ be the circumradius and inradius of the triangle,respectively. If $R:r = m:n$,where $m$ and $n$ are coprime positive integers,then $m + n$ is

In $\triangle ABC$,if $\frac{1}{b+c} + \frac{1}{c+a} = \frac{3}{a+b+c}$,then $\angle C$ is equal to: (in $^{\circ}$)

In a triangle $ABC$,with usual notations,if $c=4$,then the value of $(a-b)^2 \cos^2 \frac{C}{2} + (a+b)^2 \sin^2 \frac{C}{2}$ is

In $\triangle ABC$,if $a=13, b=14$ and $\cos \frac{C}{2}=\frac{3}{\sqrt{13}}$,then $2r_1=$