In $\triangle ABC$,if $(a-b)(s-c)=(b-c)(s-a)$,then $r_1, r_2, r_3$ are in

In a triangle,the lengths of the sides are integers. Suppose that the length of one side is $1$,and the longest altitude is twice the shortest altitude. Let $R$ and $r$ be the circumradius and inradius of the triangle,respectively. If $R:r = m:n$,where $m$ and $n$ are coprime positive integers,then $m + n$ is

In $\triangle ABC$,if $a, b, c$ are its sides and $\angle C = 60^{\circ}$,find the value of $\frac{a}{b+c} + \frac{b}{c+a}$.

With usual notations,if the angles of a triangle are in the ratio $1: 2: 3$,then their corresponding sides are in the ratio.

The sides of a triangle are in the ratio $1 : \sqrt{3} : 2$. Then the angles are in the ratio

In a triangle $ABC$,if $(b+c)^2 \sin^2\left(\frac{A}{2}\right) + (b-c)^2 \cos^2\left(\frac{A}{2}\right) = K(1 - \cos 2A)$,then $K =$