In a $\triangle ABC$,the expression $(a-b)^2 \cos^2 \frac{C}{2} + (a+b)^2 \sin^2 \frac{C}{2}$ is equal to:

If $2$ is the length of a side of a triangle with its opposite angle $\frac{\pi}{3}$,then the circumradius of the triangle is ......

The lengths of the sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the lengths of the sides of the triangle (in units) are

In a triangle $ABC,$ if $a \sin A = b \sin B,$ then the nature of the triangle is:

If the altitudes of a triangle are in arithmetic progression,then the sides of the triangle are in

In a $\Delta ABC$,if $\cos 3A + \cos 3B + \cos 3C = 1$,then: