In a triangle,if the ex-radii $r_1, r_2, r_3$ are in the ratio $1: 2: 3$,then its sides are in the ratio

The lengths of the sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the lengths of the sides of the triangle (in units) are

The number of triangles $ABC$ that can be formed with $a = 3, b = 8$ and $\sin A = \frac{5}{13}$ is

In $\triangle ABC$,if $A, B, C$ are in arithmetic progression,then $\sqrt{a^2-ac+c^2} \cdot \cos \left(\frac{A-C}{2}\right) =$

Let $ABC$ be a triangle in which $AB=BC$. Let $X$ be a point on $AB$ such that $AX:XB=AB:AX$. If $AC=AX$,then the measure of $\angle ABC$ equals (in $^{\circ}$)

In $\triangle ABC$,if $a=3, b=4, c=6$,then $\frac{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}{\cot A+\cot B+\cot C}=$