In a $\triangle ABC$,$2A + C = 300^{\circ}$. If the circumradius $R$ of the $\triangle ABC$ is eight times its inradius $r$,then $\sin \frac{C}{2} = $

In a triangle $XYZ$,let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$,respectively,and $2s = x+y+z$. If $\frac{s-x}{4} = \frac{s-y}{3} = \frac{s-z}{2}$ and the area of the incircle of the triangle $XYZ$ is $\frac{8\pi}{3}$,then:

$AD, BE$ and $CF$ are the perpendiculars from the vertices of a $\Delta ABC$ to the opposite sides. The ratio of the perimeter of $\Delta DEF$ to the perimeter of $\Delta ABC$ is: (where $r$ is the inradius and $R$ is the circumradius of $\Delta ABC$)

In a triangle $ABC$, $a=5$, $b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then its inradius $r=$

The area of the equilateral triangle which contains three coins of unit radius is

In $\triangle ABC$,if $r$ is the inradius and $r_1, r_2, r_3$ are the ex-radii,then $\frac{1}{4}[b^2 \sin 2C + c^2 \sin 2B] =$