In a $\triangle ABC$,if $a=13, b=14, c=15$,then find the value of $r_1$.

In a triangle $ABC$, $(r_2+r_3) \operatorname{cosec}^2\left(\frac{A}{2}\right) =$

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

The circumradius of triangle $PRS$ is . . . . .

In a $\triangle ABC$,let $\angle C = \frac{\pi}{2}$. If $r$ and $R$ are the inradius and circumradius of $\triangle ABC$ respectively,then $R+r=$

In a $\triangle ABC$,$D, E$,and $F$ are the points of contact of the incircle with the sides $AB, BC$,and $CA$ respectively,such that $AD=\alpha, BE=\beta$,and $CF=\gamma$. Then $\frac{\alpha \beta \gamma}{\alpha+\beta+\gamma} = $