In $\triangle ABC$,if $\frac{1}{r_1}, \frac{1}{r_2}$ and $\frac{1}{r_3}$ are in arithmetic progression,then $r_2 : r =$

In a triangle $ABC$,$l(AB)=\sqrt{23}$ units,$l(BC)=3$ units,$l(CA)=4$ units,then $\frac{\cot A+\cot C}{\cot B}$ is

If in $\triangle ABC$,$B=45^{\circ}$,$a=2(\sqrt{3}+1)$ and the area of $\triangle ABC$ is $6+2\sqrt{3}$ sq. units,then the side $b=$

With usual notations,in triangle $ABC$,$a=\sqrt{3}+1$,$b=\sqrt{3}-1$ and $m \angle C=60^{\circ}$,then $A-B=$ (in $^{\circ}$)

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 any triangle $ABC$,$(a + b)^2 \sin^2 \frac{C}{2} + (a - b)^2 \cos^2 \frac{C}{2} =$