For a triangle with side lengths $6$,$5$,and $9$,find the inradius of the triangle.

Let $ABC$ be a scalene triangle with incentre $I$ and circumcentre $O$. Suppose $B, C, I, O$ are concyclic points. Then $\angle B + \angle C$ is (in $^{\circ}$)

In $\triangle ABC$,if $b=2, c=\sqrt{3}$,and $A=30^{\circ}$,then its inradius $r=$

In a triangle $ABC$, with usual notation, if $a=12, b=16, c=20$, then the ratio of the exradii of the triangle opposite to the angles in the order $\angle C, \angle B, \angle A$ is

In triangle $ABC$,if $a=6, b=8$ and $c=10$,then $\frac{2 r_2 r_3}{r r_1} = $

$AD, BE$ and $CF$ are the perpendiculars from the vertices of a $\Delta ABC$ to the opposite sides. The ratio of the perimeter of $\Delta DEF$ to the perimeter of $\Delta ABC$ is: (where $r$ is the inradius and $R$ is the circumradius of $\Delta ABC$)