If a triangle has sides $a, b, c$ and $r_1 > r_2 > r_3$ (where $r_1, r_2, r_3$ are the ex-radii),then:

In a $\triangle ABC$,if $2 \Delta^2 = \frac{a^2 b^2 c^2}{a^2+b^2+c^2}$,then the triangle is

In a $\triangle ABC$,if $a+c=5b$,then $\cot \frac{A}{2} \cot \frac{C}{2} =$

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

With usual notation in a $\Delta ABC$,if $R = k \frac{(r_1 + r_2)(r_2 + r_3)(r_3 + r_1)}{r_1 r_2 + r_2 r_3 + r_3 r_1}$,where $k$ has the value equal to: