Inradius, Exradii and Circumradius

The value of $\frac{1}{bc}+\frac{1}{ca}+\frac{1}{ab}$ is equal to: $\{$where $a,b,c$ denote $3$ sides of a $\Delta ABC$ and $r$ is a in-radius of $\Delta ABC$ and $R$is a circum-radius of $\Delta ABC\}$

$\frac{1}{{\text{r}}^2}+\frac{1}{{\text{r}}_1^2}+\frac{1}{{\text{r}}_2^2}+\frac{1}{{\text{r}}_3^2}=$

In a triangle  $ABC,a:b:c=4:5:6.$ The ratio of the radius of the circumcircle to that of the incircle is: $\{ \mathrm{where} a=BC, b=CA, c=AB\}$

What are circumcircle and circumradius of a triangle? What is the circumradius of the equilateral triangle of sides $8$ centimetres?

What are Incircle and inradius of a triangle? Peter calculated the area of a triangular sheet as $90 {\text{feet}}^2$. The perimeter of the sheet is $30 \text{feet}$. If a circle is drawn inside the triangle such that it is touching every side of the triangle, help Peter calculate the inradius of the triangle.

In a triangle $ABC$, a point $D$ is chosen on $BC$ such that $BD:DC=2:5$. Let $P$ be a point on the circumcircle $ABC$ such that $\angle PDB=\angle BAC$. Then $PD:PC$ is :-

In $△ABC$, $r_1+r_2+r_3=$

In $∆ABC,$ the circle that touches the sides $BC$ internally and other two sides $AB$ and $AC$ externally, is called

Tangents that are drawn from the point $P(3,4)$ to the circle $\text{S}\equiv {\text{x}}^2+{\text{y}}^2-9=0$ that meets the circle at $A \& B$ . If for circle $S_1=0, S=0$ is the director circle and for the circle $S_2=0$ , its succeeding circle. Suppose, if the $\Delta \text{PAB}$ is called as tangent triangle of the circle $S=0$ . Let$A_1,B_1 ; A_2,B_{2 }; A_3,B_3,......$ be the points of contacts of tangents drawn from $P$ to the circles$S_1=0, S_2=0, ......... \& P{A}_1{B}_1, P{A}_2{B}_2, P{A}_3{B}_3,......$ be the corresponding tangent triangles. Here, director circle is the locus of point of intersection of perpendicular tangents with respect to a (circle).

 If $R_1,R_2,R_3,..........,R_n$  are the circumradii of the tangent triangles $P{A}_1{B}_1,P{A}_2{B}_2,P{A}_3{B}_3,.........,P{A}_n{B}_n$ , then $\sum _{i=1}^{i=20}i{R}_i$ is equal to

For next two question please follow the same 

Tangents that are drawn from the point $P(3,4)$ to the circle $S\equiv x^2+y^2-9=0$ that meets the circle at $A \& B$ . If for circle $S_1=0, S=0$ is the director circle and for the circle $S_2=0$ , its succeeding circle. Suppose, if the $\Delta PAB$ is called as tangent triangle of the circle $S=0$ . Let$A_1,B_1 ; A_2,B_{2 }; A_3,B_3,..$ be the points of contacts of tangents drawn from $P$ to the circles$S_1=0, S_2=0, ... \& P{A}_1{B}_1, P{A}_2{B}_2, P{A}_3{B}_3,...$ be the corresponding tangent triangles. Here, director circle is the locus of point of intersection of perpendicular tangents with respect to a (circle).

 The inradius ( in units ) of the $\Delta PAB$ is

If two sides of a triangle are $5$ and $8$, and its circum-radius is $\frac{25}{6}$, then the third side can be-

$\frac{{\mathrm{r}}_1^{\text{'}}}{\mathrm{R}}=$

$\frac{{\mathrm{r}}^{\text{'}}}{\mathrm{R}}=$

$\frac{s^{\text{'}}}{R}=$

The inradius $\mathrm{r}$ is given by

If $\frac{\mathrm{bx}}{\mathrm{c}}+\frac{\mathrm{cy}}{\mathrm{a}}+\frac{\mathrm{az}}{\mathrm{b}}=\frac{{\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2}{\mathrm{k}}$, then the value of $\mathrm{k}$ is 

Let $x$ and $y$ represent the sum and product of two sides of a triangle such that $x^2=y+z^2$ where $z$ is the third side, then inradius of the triangle is

The inradius of the triangle having sides $\mathrm{26,\; 28,\; 30}$ units, is

Circumcentre is at origin and $a \le \sin A$,  $P\left(x, y\right)$  lie inside the circumcircle & $k=\frac{1}{8\left|xy\right|}$, then least integer value of k can be

In a $\mathrm{\Delta }\mathrm{ }\mathrm{A}\mathrm{B}\mathrm{C}$ , if $\mathrm{r}\mathrm{ }=\mathrm{ }{\mathrm{r}}_2\mathrm{ }+\mathrm{ }{\mathrm{r}}_3-\mathrm{ }{\mathrm{r}}_1$ , and $\mathrm{\angle }\mathrm{A}\mathrm{ }>\frac{\mathrm{\pi }}{3}$ then the range of $\frac{\mathrm{s}}{\mathrm{a}}$ is equal to