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

In $\triangle ABC$,suppose the radius of the excircle opposite to angle $A$ is denoted by $r_1$,similarly $r_2$ for angle $B$,and $r_3$ for angle $C$. If $r$ is the radius of the inscribed circle,then what is the value of $\frac{ab - r_1 r_2}{r_3}$?

In a $\triangle ABC$,$\frac{\Delta^2}{a^2+b^2+c^2}\left(\frac{1}{r_1^2}+\frac{1}{r_2^2}+\frac{1}{r_3^2}+\frac{1}{r^2}\right) = $

Let $ABC$ be a triangle right-angled at $B$. If $a = 13$ and $c = 84$,then $r + R = $

In a triangle $ABC$,if $b = 2$ and $B = 30^\circ$,then the area of the circumcircle of triangle $ABC$ in square units is:

If the angular bisector of the angle $A$ of the triangle $ABC$ meets its circumcircle at $E$ and the opposite side $BC$ at $D$,then $DE \cos \frac{A}{2} = $