In $\triangle ABC$,if $a:b:c = 4:5:6$,then the ratio of the circumradius to its inradius is

In $\Delta ABC,$ the lengths of sides $AC$ and $AB$ are $12 \, cm$ and $5 \, cm,$ respectively. If the area of $\Delta ABC$ is $30 \, cm^{2}$ and $R$ and $r$ are respectively the radii of the circumcircle and incircle of $\Delta ABC,$ then the value of $2R + r$ (in $cm$) is equal to ....... .

In $\triangle ABC$,$\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2} =$

In $\triangle ABC$,if $a=7, b=8$ and $c=9$,then $\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}=$

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

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