If $A, B, C$ are the angles of a triangle,then $\frac{\sin A+\sin B+\sin C}{\sin ^2 \frac{A}{2}-\sin ^2 \frac{B}{2}+\sin ^2 \frac{C}{2}-1} =$

In a triangle $ABC$,if $\sin \frac{A}{2} = \frac{1}{4} \sqrt{\frac{3}{5}}$,$a = 2$,$c = 5$ and $b$ is an integer,then the area (in sq. units) of triangle $ABC$ is

The product of the arithmetic mean of the lengths of the sides of a triangle and the harmonic mean of the lengths of the altitudes of the triangle is equal to: [where $\Delta$ is the area of the triangle $ABC$]

Consider an obtuse-angled triangle $ABC$ in which the difference between the largest and the smallest angle is $\frac{\pi}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of radius $1$.
$(1)$ Let $a$ be the area of the triangle $ABC$. Then the value of $(64 a)^2$ is
$(2)$ The inradius of the triangle $ABC$ is

In a triangle $ABC$,if $a^2-b^2-c^2=bc(\lambda^2-2\lambda-1)$,then

In $\triangle ABC$, $A, B$ and $C$ are in arithmetic progression and $a: c = 1: 2$. If $b = 4 \sqrt{3} \text{ cm}$, then the area of $\triangle ABC$ (in $\text{sq. cm}$) is (in $\sqrt{3}$)