In triangle $ABC$,if $a=4, b=3, c=2$,then $2(a-b \cos C)(a-c \sec B) = $

In $\Delta ABC$,if $2s = a + b + c$,then the value of $\frac{s(s - a)}{bc} - \frac{(s - b)(s - c)}{bc} = $

In $\Delta ABC$,if $\tan \frac{A}{2} \tan \frac{C}{2} = \frac{1}{2}$,then $a, b, c$ are in

In a triangle $ABC$,if $r_1 = 2r_2 = 3r_3$,then $\sin A : \sin B : \sin C =$

In a triangle,the sum of lengths of two sides is $x$ and the product of the lengths of the same two sides is $y$. If $x^2 - c^2 = y$,where $c$ is the length of the third side of the triangle,then the circumradius of the triangle is

In a $\Delta ABC$,the expression $\left( \frac{a^2}{\sin A} + \frac{b^2}{\sin B} + \frac{c^2}{\sin C} \right) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ simplifies to,where $\Delta$ is the area of the triangle.