In $\triangle ABC$, find the value of $r_1+r_2+r_3$.

In $\triangle ABC$,if $a=13 \text{ cm}, b=14 \text{ cm}$ and $c=15 \text{ cm}$,then its circumradius $R$ is:

In a triangle $XYZ$,let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$,respectively,and $2s = x+y+z$. If $\frac{s-x}{4} = \frac{s-y}{3} = \frac{s-z}{2}$ and the area of the incircle of the triangle $XYZ$ is $\frac{8\pi}{3}$,then:

In $\triangle ABC$,with usual notation,observe the two statements given below :
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?

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 a $\triangle ABC$,$a: b: c = 4: 5: 6$. The ratio of the radius of the circumcircle to that of the incircle is