If in a $\Delta ABC$,$a, b, c$ are in $A.P.$,then $\tan \frac{A}{2} \tan \frac{C}{2} = $

In a triangle $ABC$ with usual notations,if $\angle A = 30^{\circ}$,then the value of $\left(1+\frac{a}{c}+\frac{b}{c}\right)\left(1+\frac{c}{b}-\frac{a}{b}\right)=$

In triangle $ABC$,if $\angle A = 45^\circ$ and $\angle B = 75^\circ$,then $a + c\sqrt{2} = $

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

The lengths of the sides of a triangle are $13$, $14$ and $15$. If $R$ and $r$ respectively denote the circumradius and inradius of that triangle, then $8R + r =$