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 $\Delta ABC\}$

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

$a·cotA+b·cotB+c·\cot C=$

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

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.

Divya inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of region between the two circles. Mansi did the same with a regular heptagon. The area of the regions calculated by Divya and Mansi are $A$ and $B$, respectively. Each polygon had a side length of $2$. Which of the following is true?

Let $ABC$ be triangle with $AB=AC=6$. If the circum radius of the triangle is $5$, then $BC$ equals

Let $A_1{A}_2{A}_3\dots \dots \dots \dots .A_9$ be a nine-sided regular polygon with side length $2$ units. The difference between the lengths of the diagonals $A_1{A}_5$ and $A_2{A}_4$ equals

If $R$ is the circum radius of $\Delta ABC$ , then $A\left(\Delta ABC\right)$ = ….

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 $\Delta ABC\}$

In $\text{Δ}\text{ABC}$ , BC = 13 cm, AC = 14 cm and AB = 15 cm, then its circum-radius is equal to

$AD, BE, CF$ are internal angular bisectors of $∆ABC$ and $I$ is the incentre. If $a(b+c)\sec \frac{A}{2}ID+b(a+c)\sec \frac{B}{2}IE+c(a+b)\sec \frac{C}{2}IF=kabc,$ then the value of $k$ is:

Let $H$ be the orthocentre of triangle $ABC.$ Then the angle subtended by side $BC$ at the centre of incircle of $∆CHB$ is:

$$  ΔABC   r is inradius and r₁, r₂, r₃ are exradii opposite to vertex A, vertex B and , vertex C respectively
If r₁ = 8, r₂ = 12, r₃ = 24, then r is equal to 

In a triangle $\left(1-\frac{r_1}{r_2}\right)\left(1-\frac{r_1}{r_3}\right)=2$, then the triangle is ( r₁, r₂ and r₃ are exradii of traingle)

In any $∆ABC,$ the line joining the circumcentre $\left(O\right)$ and incentre $\left(I\right)$ is parallel to $AC,$ then $OI$ is equal to-

Find the area of the circumcircle of $∆ABC$, if $b=2, \angle B=30°$.

In a triangle, if $r_1 > r_{2 }> r_3$.Tick the appropriate option.

Let $a, b,$ and $c$ be the side lengths of a triangle $ABC$ and assume that $a\le b$ and $a\le c.$ If $x=\frac{b+c-a}{2},$ then find the minimum value of $\frac{ax}{rR},$ where $r$ and $R$ denote inradius and circumradius of triangle $ABC.$

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.