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:

In $\triangle ABC$,if $A$ is an acute angle,$b=6, c=9$ and $\sin A=\frac{2 \sqrt{14}}{9}$,then $3a(\cos B+\cos C)=$

In a $\triangle ABC$,if $a^4+b^4+c^4=2b^2c^2+2a^2b^2$,then $B=$

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 a $\triangle ABC$,$\sin 2A + \sin 2B + \sin 2C =$

If $R$ is the circumradius of $\triangle ABC$,then the area of $\triangle ABC$ is equal to $\ldots$.