In a triangle $ABC$,if $a < b < c$ and $\frac{a^3+b^3+c^3}{\sin^3 A+\sin^3 B+\sin^3 C}=8$,then the maximum value of $c$ is

In $\Delta ABC$,$a \cot A + b \cot B + c \cot C = . . . $ (where $r$ is inradius and $R$ is circumradius.)

The sides $a, b, c$ of a triangle satisfy the relations $c^2=2ab$ and $a^2+c^2=3b^2$. Then the measure of $\angle BAC$,in degrees,is

In an isosceles triangle $ABC$,$\angle C = \angle A$. If the point of intersection of the bisectors of internal angles $\angle A$ and $\angle C$ divides the median of side $AC$ in the ratio $3 : 1$ (from vertex $B$ to side $AC$),then the value of $\csc \frac{B}{2}$ is equal to

In $\triangle ABC$,if $a, b$ and $c$ are in arithmetic progression,then $\cos A + 2 \cos B + \cos C =$

If $R = \frac{65}{8}$,$r_1 = \frac{21}{2}$,and $r_2 = 12$ are the circumradius and the radii of the excircles opposite to the vertices $A$ and $B$ of a triangle $ABC$ respectively,then the area of the triangle (in square units) is