Inradius, Exradii and Circumradius

A triangle and a parallelogram are constructed on the same base such that their areas are equal. If the altitude of the parallelogram is $100 \mathrm{m}$, then the altitude of the triangle is?

In $∆\mathrm{ABC}$, $\angle A<\angle B$, the altitude to the base divides vertex angle $C$ into two parts $C_1$ and $C_2$ adjacent to $BC$, then which one of the relation can be obtained?

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

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)

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.

$ABCD$ is a cyclic quadrilateral. Prove the result:

$AC\left[AB·BC+CD·DA\right]=BD\left[AB·AD+CB·CD\right]$

What is Euler line of a triangle?

If the radius of a nine point circle is $10 \mathrm{cm}$, find the radius (in $\mathrm{cm}$) of circumcircle of the triangle.

The nine point circleof triangle $∆ABC$ is shown. Find $\angle ALB$ in degrees?

Nine point circle of any triangle is also known as _____

The nine point circle of equilateral triangle $∆XYZ$ is shown. If $XP=6 \mathrm{cm}$, find the length of $XZ$(in $\mathrm{cm}$)?

Define nine point circle.

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 the length of the two altitudes of a $∆PQR$ are $9$$cm$ and $12$$cm$, then what would be the possible length of the unknown altitude?

If in a triangle $\left(1-\frac{r_1}{r_2}\right)\left(1-\frac{r_1}{r_3}\right)=2,$ then the triangle is