In a $\triangle A B C$, if $(b+c)^2 \sin ^2 \frac{A}{2}+(b-c)^2 \cos ^2 \frac{A}{2}=K(1-\cos 2 A)$, then $K=$

In a $\triangle A B C$, if $b=7, c=4 \sqrt{3}$ and $A=\frac{\pi}{6}$ then a $\sin B \sin C=$

In $\triangle A B C$, if $B C$ is the hypotenuse, then $r_2+r_3=$

If the sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the area (in sq. units) of that triangle is