In $\triangle ABC$, with usual notations, if the sides $a, b$ and $c$ are in the ratio $18 : 17 : 7$, then $\cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = $

In $\Delta ABC$,$(a-b)^2 \cos^2 \frac{C}{2} + (a+b)^2 \sin^2 \frac{C}{2} =$

Given $\frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13}$ for a $\Delta ABC$ with usual notation. If $\frac{\cos A}{\alpha} = \frac{\cos B}{\beta} = \frac{\cos C}{\gamma}$,then the ordered triple $(\alpha, \beta, \gamma)$ is proportional to

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 $8R + r =$

In triangle $ABC,$ $\frac{1 + \cos(A - B)\cos C}{1 + \cos(A - C)\cos B} = $

Let $ABC$ be an isosceles triangle with $BC$ as its base. Then, $r_1=$