If $A = 30^\circ, c = 7\sqrt{3}$ and $C = 90^\circ$ in $\Delta ABC$,then $a = $

If ${p_1}, {p_2}, {p_3}$ are altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively and $\Delta$ is the area of the triangle,then ${p_1}^{-2} + {p_2}^{-2} + {p_3}^{-2}$ is equal to

In any triangle $ABC$,$(b+c) \cos A + (c+a) \cos B + (a+b) \cos C$ equals

In $\triangle ABC$,with usual notations,$2ab \sin \frac{1}{2}(A+B-C) =$

In any $\triangle ABC$,$\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} =$

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