If the lengths of the sides of a triangle are $3, 4$ and $5$ units,then $R$ (the circumradius) is ............ $unit$.

In an equilateral triangle,the inradius $(r)$ and the circumradius $(R)$ are connected by which of the following relations?

$ABC$ 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

In $\triangle ABC$,find the value of $\frac{r_1-r}{a}+\frac{r_2-r}{b}$.

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

If $d_1, d_2, d_3$ are the diameters of three ex-circles of a $\triangle ABC$,then $d_1 d_2 + d_2 d_3 + d_3 d_1 =$