Inradius, Exradii and Circum-radius

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 $\Delta ABC\}$

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

$a·cotA+b·cotB+c·\cot C=$

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

In $\mathit{∆}ABC\mathit{,}$ the ratio $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ is not always equal to (All symbols used have usual meaning in a triangle)

For a regular polygon, let '$r$' and '$R$' be the radii of the inscribed and the circumscribed circles respectively. Which of the following is/are true

In triangle $ABC$, Let $a,b,c$ denote the lengths of the sides opposite to the vertices $A,B$ and $C$ respectively. If $a,b,c$ are in arithmetic progression such that $x^2+3x+5=0$ and $3x^2+ax+c=0$ have a common root, then the radius of the smallest circle which touch all the sides of triangle $ABC$ is

Prove that in any $\Delta ABC, R+r\ge \min \left({\ell }_a,{\ell }_b,{\ell }_c\right)$, where $R$ is the circumradius, $r$ the inradius, and ${\ell }_a,{\ell }_b,{\ell }_c$ the angle bisectors of the triangle.

If $\left(1-\frac{r_1}{r_2}\right)\left(1-\frac{r_1}{r_3}\right)=2$, then prove that the triangle is right-angled.

In any $∆ABC$, prove that

(i) $\left(r_3+r_1\right)\left(r_3+r_2\right)\sin C=2r_3\sqrt{r_2{r}_3+r_3{r}_1+r_1{r}_2}$.

In an acute-angled triangle $ABC, r+r_1=r_2+r_3$ and $\angle B>\frac{\mathrm{\pi }}{3}$, then prove that $b+3c<3a<3b+3c$.

Let $ABC$ be a triangle with $\angle BAC=\frac{2\pi }{3}$ and $AB=x$ such that $\left(AB\right)\left(AC\right)=1$. If $x$ varies, then the longest possible length of the internal angle bisector $AD$ equals

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 a triangle $ABC$, the medians $AD, BE$ and $CF$ pass through the point $G$. If $AD=9 \mathrm{cm}, GE=4.2 \mathrm{cm}$  and $GC=6 \mathrm{cm}$, then the values of the lengths of $AG\mathit{,}\mathit{ }BE$ and $FG$ respectively are:

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

In a triangle $ABC, AD$ is the altitude from $A.$ Given $b>c, \angle C=23°$ and $AD=\frac{abc}{b^2-c^2},$ then $\angle B=$

If $\mathrm{l}$ has greatest possible integer value then $∆\mathrm{ABC}$ is