In a triangle,if $b=5, c=6$,and $\tan \frac{A}{2}=\frac{1}{\sqrt{2}}$,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 a triangle $ABC$,let $AB = \sqrt{23}$,$BC = 3$,and $CA = 4$. Then the value of $\frac{\cot A + \cot C}{\cot B}$ is:

In a $\triangle ABC$,the sides $a, b, c$ are in $A$.$P$. if and only if $r_1, r_2, r_3$ are in

In a $\Delta ABC$,$a = 5, b = 4$ and $\cos(A - B) = \frac{31}{32}$,then side $c$ is equal to

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