In a $\triangle ABC$, if $a: b: c = 4: 5: 6$, then the ratio of the radius of its circumcircle to that of its incircle is:

If the lengths of the sides of a triangle are $15, 20, 25$ units,find the circumradius of the triangle. (in $units$)

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} = $

In $\triangle ABC$,suppose the exradii opposite to angles $A, B$ and $C$ are denoted by $r_1, r_2$ and $r_3$ respectively. If $r_1=2, r_2=3, r_3=6$ and $R$ is the radius of the circumcircle,then the value of $r_1+r_2+r_3-r$ is:

In a $\triangle ABC$,let $\angle C = \frac{\pi}{2}$. If $r$ and $R$ are the inradius and circumradius of $\triangle ABC$ respectively,then $R+r=$

If in a $\triangle ABC$, $AD$, $BE$, and $CF$ are the altitudes and $R$ is the circumradius of $\triangle ABC$, then the radius of the circumcircle of $\triangle DEF$ is