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

In $\triangle ABC$,if $r_1=3, r_2=10$ and $r_3=15$,then $R=$

$ABC$ is an isosceles triangle with an inscribed circle with center $O$. Let $P$ be the midpoint of $BC$. If $AB = AC = 15$ and $BC = 10$, then $OP$ equals

Let $ABC$ be an equilateral triangle with side length $a$. Let $R$ and $r$ denote the radii of the circumcircle and the incircle of triangle $ABC$ respectively. Then,as a function of $a$,the ratio $\frac{R}{r}$

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 $\triangle ABC$ is a right-angled isosceles triangle and $\angle C = 90^{\circ}$,then $r : r_3 =$