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: $\{ \mathrm{where} a=BC, b=CA, c=AB\}$

In a $△ABC$, If $\frac{r_1}{bc}+\frac{r_2}{ca}+\frac{r_3}{ab}=\frac{1}{r}-\frac{1}{nR}$ then the value of $n$ is

If the length of each side of a regular pentagon is equal to $6 m$ then find the length of its inradius. [Use $\cot \left(36°\right)=1.3764$].

If the length of each side of a regular pentagon is equal to $6 m$ then find the length of its circum-radius.  [Use $\mathrm{cosec}\left(36°\right)=1.70130162$]

If the length of each side of a regular pentagon is equal to $3 m$ then find the length of its inradius.

If the length of each side of a regular pentagon is equal to $3 m$ then find the length of its circum-radius.

The lengths of the tangents drawn from the vertices $\mathrm{A}, \mathrm{B}, \mathrm{C}$ to the incircle of $\mathrm{\Delta ABC}$ are $5, 3, 2$ respectively. If the lengths of the parts of tangents with in the triangle which are drawn parallel to the sides $\mathrm{BC}, \mathrm{CA}, \mathrm{AB}$ of the triangle to the incircle be $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }$ respectively, then $[\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }]$ (where $[\bullet ]$ is G.I.F. is)

Mention the formulae to find the in-radius (Apothem) of a regular polygon, when its (a) side length is given (b) circum-radius is given

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 $R\left(a+b\right)=c\sqrt{ab}$, where $a,b,c$ are sides of $∆ABC$ and $R$ is circumradius, then $\cos \left(\frac{C}{2}\right)\sin \left(\frac{C}{3}\right)=\_\_\_\_\_$

The Circumcenter of a triangle is the intersection point of which of the following?

Define Circum-circle of a Triangle. Find Circum-radius, when $a=4 \text{cm},b=5 \text{cm},c=6 \text{cm}$ and $∆=6 {\text{cm}}^2$. [Write your answer excluding units]

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

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.