In $\Delta ABC,$ if $\sin^2 \frac{A}{2}, \sin^2 \frac{B}{2}, \sin^2 \frac{C}{2}$ are in $H.P.,$ then $a, b, c$ will be in

In a triangle $ABC$,$CH$ and $CM$ are the lengths of the altitude and median to the base $AB$ respectively. If $a = 10, b = 26, c = 32$,then find the length of $HM$.

If $r_1, r_2$ and $r_3$ of a $\triangle ABC$ are in Harmonic progression,then $a, b$ and $c$ will be in

In $\Delta ABC$,the expression $a(\cos^2 B + \cos^2 C) + \cos A(c \cos C + b \cos B)$ is equal to:

In a $\triangle ABC$,if $\cos A + \cos B + \cos C = a + b \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$,then $(a + b)$ is equal to

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