The area (in square units) of $\triangle ABC$ if $\angle A=75^{\circ}, \angle B=45^{\circ}$ and $a=2(\sqrt{3}+1)$ is

In triangle $ABC$,if $A$ is acute,$C$ is obtuse,$\sin A = \frac{3\sqrt{3}}{14}$,$a = 3$,and $b = 5$,then $c =$

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 $\frac{1}{a+b} + \frac{1}{c+a} = \frac{3}{a+b+c}$,then $\sin A$ is equal to

In triangle $ABC$,$(b + c)\cos A + (c + a)\cos B + (a + b)\cos C = $

In a triangle $ABC$,$\cos 3A + \cos 3B + \cos 3C = 1$. If the circumradius of triangle $ABC$ is $\sqrt{3}$,then the length of its longest side is