In a triangle $ABC$,if $b = 2$ and $B = 30^\circ$,then the area of the circumcircle of triangle $ABC$ in square units is:

In an equilateral triangle,the inradius $(r)$ and the circumradius $(R)$ are connected by which of the following relations?

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 $\triangle ABC$,with usual notation,observe the two statements given below :
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?

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} = $

If $a, b, c$ are the sides of a $\Delta ABC$ and exradii $r_1, r_2, r_3$ are respectively $12, 6, 4$,then $a+2b+3c=$