In a $\Delta ABC$,if $2s = a + b + c$ and $(s - b)(s - c) = x \sin^2 \frac{A}{2}$,then $x =$

If $a, b, c$ are the sides and $A, B, C$ are the angles of a triangle $ABC$,then $\tan \left( \frac{A}{2} \right)$ is equal to

In $\Delta ABC$,$({b^2} - {c^2})\cot A + ({c^2} - {a^2})\cot B + ({a^2} - {b^2})\cot C = $

Consider a triangle $ABC$ and let $a, b$ and $c$ denote the lengths of the sides opposite to vertices $A, B$ and $C$ respectively. Suppose $a=6, b=10$ and the area of the triangle is $15 \sqrt{3}$. If $\angle ACB$ is obtuse and if $r$ denotes the radius of the incircle of the triangle,then $r^2$ is equal to

In $\triangle ABC$,if $\frac{1}{b+c} + \frac{1}{c+a} = \frac{3}{a+b+c}$,then $\angle C$ is equal to: (in $^{\circ}$)

The inradius of $\Delta ABC$ is $3$ units and the exradius opposite to vertex $A$ is $4$ units. Then,the length of the altitude from vertex $A$ is ............. $units$.