If $\triangle ABC$ is a right-angled isosceles triangle and $\angle C = 90^{\circ}$,then $r : r_3 =$

In triangle $ABC$, if $b=6, c=7$ and $\tan \frac{A}{2}=\frac{1}{\sqrt{6}}$, then the inradius of $\triangle ABC$ is

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} = $

In a $\triangle ABC$,which of the following formulae are correct?
$I. r = 4R \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$
$II. r_1 = (s-a) \tan \frac{A}{2}$
$III. r_3 = \frac{\Delta}{s-c}$

In $\triangle ABC$,if $2R + r = r_2$,then $\angle B$ is equal to

The inradius of a circle inscribed in an isosceles right-angled triangle,where one of the angles is $\frac{\pi}{2}$,is $1$. Find the area of the triangle (in sq. units).