Half-Angle Formula and the Area of a Triangle

In a quadrilateral $ABCD,$ it is given that $AB=AD=13, BC=CD=20, BD=24 .$ If $r$ is the radius of the circle inscribed in the quadrilateral, then the integer closest to $r$ is

In a $∆ABC, X$ and $Y$ are points on the segment $AB$ and $AC$ respectively, such that $AX:XB=1:2$ and $AY:YC=2:1$. If the area of $∆AXY$ is $10$ sq. units, then the area of $∆ABC$ in sq. units, is

Let $ABCD$ be a square and $E$ be a point outside $ABCD$ such that $E, A, C$ are collinear in that order. Suppose $EB=ED=\sqrt{130}$ and the areas of triangle $EAB$ and square $ABCD$ are equal. Then the area of square $ABCD$ is :

A triangle with perimeter $7$ has integer side lengths. What is the maximum possible area of such a triangle?

If $\mathrm{p},\mathrm{ }\mathrm{q},\mathrm{ }\mathrm{r}$ are the lengths of the internal bisectors of angles $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C}$ of a $\mathrm{\Delta }\mathrm{A}\mathrm{B}\mathrm{C}$ respectively, then $\frac{1}{\mathrm{p}}\cos \frac{\mathrm{A}}{2}+\frac{1}{\mathrm{q}}\cos \frac{\mathrm{B}}{2}+\frac{1}{\mathrm{r}}\cos \frac{\mathrm{C}}{2}$ is equal to ( where a = BC, b = CA, c = AB)

If $p_1, p_2, p_3$ are respectively the length of the perpendicular from the vertices of a $∆ABC$ to the opposite sides, then $p_1{p}_2{p}_3$ is equal to

(where $a=BC, b=AC, c=AB \& R=\text{circumradius of }∆ABC$) 

If in a triangle $a{\cos }^2\left(\frac{C}{2}\right)+c{\cos }^2\left(\frac{A}{2}\right)=\frac{3b}{2},$ then the sides of the triangle are in:

If the length of each side of an equilateral triangle is 10cm, then its area is

If in any triangle, the area of the triangle $∆\le \frac{b^2+c^2}{\lambda },$ then the largest possible numerical value of $\lambda$ is:

The diagonals of a convex quadrilateral intersect in $O.$ What is the smallest area this quadrilateral can have, if the triangles $AOB$ and $COD$ have areas $4$ and $9,$ respectively ?

If $\mathrm{h}$ is the perpendicular distance from $\mathrm{A}$ on $\mathrm{BC}$ of a triangle $\mathrm{ABC}$, prove that $h=\frac{a\mathrm{sinBsin}C}{\sin (\mathrm{B}+\mathrm{C})}$.

[x] $(b-c)\cot \frac{A}{2}+(c-a)\cot \frac{B}{2}+(a-b)\cot \frac{C}{2}=0$.

[ix]$\frac{b-c}{b}{\cos }^2\frac{\mathrm{A}}{2}+\frac{c-a}{b}{\cos }^2\frac{\mathrm{B}}{2}+\frac{a-b}{c}{\cos }^2\frac{\mathrm{C}}{2}=0$.

In a $△ABC, \tan \frac{A}{2}=\frac{5}{6}, \tan \frac{B}{2}=\frac{20}{37}$; prove that, $a+c=2b$.

In a right-angled triangle $ABC$, right angled at $A$, let $l$ be the incentre, If $IB=\sqrt{10}$ units and $IC=\sqrt{5}$ units, find the area of $△ABC$.

If $x, y$ and $z$ be the lengths of the perpendiculars from circum centre of a triangle $ABC$ on the sides $\overline{BC},\overline{ CA}$ and $\overline{AB}$ respectively, then show that
$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{abc}{4xyz}$.

Let $ABC$ be a triangle such that $AB=4,BC=5$ and $CA=6$. Choose points $D,E,F$ on $AB,BC,CA$ respectively, such that $AD=2,BE=3,CF=4$. Then $\frac{\mathrm{area} \mathrm{\Delta }DEF}{\mathrm{area} \mathrm{\Delta }ABC}$ is

Let $P$ be a point in the interior of the rectangle $ABCD$ Which of the following sets of numbers can form the areas of the four triangles $PAB, PBC, PCD, PDA$ in same order?

Show that in $∆ABC$, $b{\cos }^2\frac{C}{2}+c{\cos }^2\frac{B}{2}=s$.

Show that in $∆ABC$ $\left(\tan \frac{A}{2}\right)\left(\tan \frac{B}{2}\right)=\frac{a+b-c}{a+b+c}$.