Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Problem 6

$a(\cos C-\cos B)=2(b-c) \cos ^{2} \frac{A}{2}$

Aman Gupta

Numerade Educator

01:50

Problem 7

$\frac{\sin (B-C)}{\sin (B+C)}=\frac{b^{2}-c^{2}}{a^{2}}$

Aman Gupta

Numerade Educator

Problem 8

$\frac{a+b}{a-b}=\tan \frac{A+B}{2} \cot \frac{A-B}{2}$

Aman Gupta

Numerade Educator

Problem 9

$a \sin \left(\frac{A}{2}+B\right)=(b+c) \sin \frac{A}{2}=a \cos \left(\frac{B-C}{2}\right)$.

Aman Gupta

Numerade Educator

02:00

Problem 10

$\frac{a^{2} \sin (B-C)}{\sin B+\sin C}+\frac{b^{2} \sin (C-A)}{\sin C+\sin A}+\frac{c^{2} \sin (A-B)}{\sin A+\sin B}=0 .$

Aman Gupta

Numerade Educator

01:53

Problem 11

$(b+c-a)\left(\cot \frac{B}{2}+\cot \frac{C}{2}\right)=2 a \cot \frac{A}{2}$.

Aman Gupta

Numerade Educator

01:06

Problem 12

$a^{2}+b^{2}+c^{2}=2(b c \cos A+c a \cos B+a b \cos C)$

Aman Gupta

Numerade Educator

Problem 13

$\left(-a^{2}+b^{2}+c^{2}\right) \tan A=\left(a^{2}-b^{2}+c^{2}\right) \tan B=\left(a^{2}+b^{2}-c^{2}\right) \tan C .$

Aman Gupta

Numerade Educator

01:20

Problem 14

$c^{2}=(a-b)^{2} \cos ^{2} \frac{C}{2}+(a+b)^{2} \sin ^{2} \frac{C}{2}$.

Aman Gupta

Numerade Educator

02:59

Problem 15

$a \sin (B-C)+b \sin (C-A)+c \sin (A-B)=0 .$

Aman Gupta

Numerade Educator

Problem 16

$\frac{a \sin (B-C)}{b^{2}-c^{2}}=\frac{b \sin (C-A)}{c^{2}-a^{2}}=\frac{c \sin (A-B)}{a^{2}-b^{2}} .$

Aman Gupta

Numerade Educator

Problem 17

$a \sin \frac{A}{2} \sin \frac{B-C}{2}+b \sin \frac{B}{2} \sin \frac{C-A}{2}+c \sin \frac{C}{2} \sin \frac{A-B}{2}=0 .$

Aman Gupta

Numerade Educator

Problem 18

$a^{2}\left(\cos ^{2} B-\cos ^{2} C\right)+b^{2}\left(\cos ^{2} C-\cos ^{2} A\right)+c^{2}\left(\cos ^{2} A-\cos ^{2} B\right)=0 .$

Aman Gupta

Numerade Educator

03:52

Problem 19

$\frac{b^{2}-c^{2}}{a^{2}} \sin 2 A+\frac{c^{2}-a^{2}}{b^{2}} \sin 2 B+\frac{a^{2}-b^{2}}{c^{2}} \sin 2 C=0 .$

Aman Gupta

Numerade Educator

01:48

Problem 20

$\frac{(a+b+c)^{2}}{a^{2}+b^{2}+c^{2}}=\frac{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}}{\cot A+\cot B+\cot C}$

Aman Gupta

Numerade Educator

Problem 21

$a^{3} \cos (B-C)+b^{3} \cos (C-A)+c^{3} \cos (A-B)=3 a b c$

Aman Gupta

Numerade Educator

Problem 22

$a^{2} \cos 2 B+b^{2} \cos 2 A+2 a b \cos (A-B)=c^{2}$

Aman Gupta

Numerade Educator

Problem 23

$(a+b+c)(\cos A+\cos B+\cos C)=2\left(a \cos ^{2} \frac{A}{2}+b \cos ^{2} \frac{B}{2}+c \cos ^{2} \frac{C}{2}\right)$

Aman Gupta

Numerade Educator

03:24

Problem 24

$\left(b^{2}-c^{2}\right) \cot A+\left(c^{2}-a^{2}\right) \cot B+\left(a^{2}-b^{2}\right) \cot C=0 .$

Aman Gupta

Numerade Educator

Problem 25

$a^{2}=(b-c)^{2}+4 b c \sin ^{2} \frac{A}{2}$

Aman Gupta

Numerade Educator

02:35

Problem 26

$\frac{1+\cos (A-B) \cos C}{1+\cos (A-C) \cos B}=\frac{a^{2}+b^{2}}{a^{2}+c^{2}}$.

Aman Gupta

Numerade Educator

02:05

Problem 27

$a(\cos B \cos C+\cos A)=b(\cos C \cos A+\cos B)=c(\cos A \cos B+\cos C)$

Aman Gupta

Numerade Educator

Problem 28

$\frac{c}{a-b}=\frac{\tan \frac{A}{2}+\tan \frac{B}{2}}{\tan \frac{A}{2}-\tan \frac{B}{2}}$

Aman Gupta

Numerade Educator

02:19

Problem 29

$\frac{c}{a+b}=\frac{1-\tan \frac{A}{2} \tan \frac{B}{2}}{1+\tan \frac{A}{2} \tan \frac{B}{2}}$

Aman Gupta

Numerade Educator

01:40

Problem 30

$\frac{a^{2} \sin B \sin C}{2 \sin A}=\Delta$.

Aman Gupta

Numerade Educator

01:39

Problem 31

$\frac{s}{\cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}}=2 \sqrt[3]{\frac{a b c}{\sin A \sin B \sin C}}$

Aman Gupta

Numerade Educator

Problem 32

$\frac{1}{(b+c)^{2}} \cos ^{2}\left(\frac{B-C}{2}\right)+\frac{1}{(b-c)^{2}} \sin ^{2}\left(\frac{B-C}{2}\right)=\frac{1}{a^{2}}$

Aman Gupta

Numerade Educator

01:27

Problem 33

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

Aman Gupta

Numerade Educator

03:16

Problem 34

$a^{2}-2 a b \cos \left(60^{\circ}+C\right)=c^{2}-2 b c \cos \left(60^{\circ}+A\right)$

Aman Gupta

Numerade Educator

09:20

Problem 35

$a^{3} \sin (B-C)+b^{3} \sin (C-A)+c^{3} \sin (A-B)=0 .$

Urvashi Arora

Numerade Educator

01:38

Problem 36

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

Aman Gupta

Numerade Educator

02:07

Problem 37

$\frac{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}{4 b^{2} c^{2}}=\sin ^{2} A$

Aman Gupta

Numerade Educator

01:17

Problem 38

$(a+b+c)\left(\tan \frac{A}{2}+\tan \frac{B}{2}\right)=2 c \cot \frac{C}{2}$.

Aman Gupta

Numerade Educator

04:03

Problem 39

$\left(\cot \frac{A}{2}+\cot \frac{B}{2}\right)\left(a \sin ^{2} \frac{B}{2}+b \sin ^{2} \frac{A}{2}\right)=c \cot \frac{C}{2}$.

Aman Gupta

Numerade Educator

01:26

Problem 40

$1-\tan \frac{A}{2} \tan \frac{B}{2}=\frac{2 c}{a+b+c} .$

Aman Gupta

Numerade Educator

Problem 41

$2 a b c \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}=2 s \Delta$

Aman Gupta

Numerade Educator

Problem 42

$b c \cos ^{2} \frac{A}{2}+c a \cos ^{2} \frac{B}{2}+a b \cos ^{2} \frac{C}{2}=s^{2}$

Aman Gupta

Numerade Educator

Problem 43

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

Aman Gupta

Numerade Educator

02:40

Problem 44

$\frac{2}{(a-b)(a-c)}+\frac{2}{(b-c)(b-a)}+\frac{2}{(c-a)(c-b)}=\frac{1}{\Delta}$

Aman Gupta

Numerade Educator

Problem 45

$\sin ^{3} A \cos (B-C)+\sin ^{3} B \cos (C-A)+\sin ^{3} C \cos (A-B)=3 \sin A \sin B \sin C .$

Aman Gupta

Numerade Educator

Problem 46

$a^{3} \cos B \cos C+b^{3} \cos C \cos A+c^{3} \cos A \cos B=a b c(1-2 \cos A \cos B \cos C)$

Aman Gupta

Numerade Educator

02:20

Solve the triangle, given
i. $\quad a=\sqrt{3}, b=\sqrt{2}$ and $c=\frac{\sqrt{6}+\sqrt{2}}{2}$.\
ii. $\quad b=\sqrt{3}, c=1$ and $A=30^{\circ} .$\
iii. $a=5, b=7$ and $A=60^{\circ}$.
iv. $a=1, c=2$ and $A=30^{\circ}$.
v. $\quad a=2, c=\sqrt{3}+1$ and $A=45^{\circ}$.=
vi. $a=\sqrt{3}, b=\sqrt{2}$ and $A=60^{\circ} .$
vii. $a=4, b=5$ and $A=120^{\circ}$.
ix. $\quad a=2, B=60^{\circ}$ and $C=45^{\circ} .$
x. $A=45^{\circ}, B=60^{\circ}$ and $C=75^{\circ} .$

Jay Patel

Numerade Educator

Problem 48

In a $\triangle A B C$, if $A=45^{\circ}, b=\sqrt{6}, a=2$, then find $B$.

Aman Gupta

Numerade Educator

01:04

Problem 49

In triangle $A B C, A=30^{\circ}, b=8, a=6$, then find $B$.

Aman Gupta

Numerade Educator

Problem 50

If $A=30^{\circ}, a=7, b=8$ in $\Delta A B C$, then how many values of $B$ are possible?

Aman Gupta

Numerade Educator

01:17

Problem 51

If the data given to construct a triangle $A B C$ are $a=5, b=7, \sin A=\frac{3}{4}$, then how many triangles can be constructed?

Aman Gupta

Numerade Educator

01:21

Problem 52

In a triangle whose sides are 3,4 and $\sqrt{38}$ meters respectively, prove that the largest angle is greater then $120^{\circ} .$

Aman Gupta

Numerade Educator

01:06

Problem 53

If in any triangle the angle be to one another as $1: 2: 3$, prove that the corresponding sides are as $1: \sqrt{3}: 2$.

Aman Gupta

Numerade Educator

03:15

Problem 54

In any triangle, if $\tan \frac{A}{2}=\frac{5}{6}$ and $\tan \frac{B}{2}=\frac{20}{37}$, find $\tan \frac{C}{2}$ and prove that in this triangle $a+c=2 b$.

Aman Gupta

Numerade Educator

01:32

Problem 55

In a $\Delta A B C, a=13, b=14, c=15$, then find $\sin \frac{A}{2}$.

Aman Gupta

Numerade Educator

02:45

Problem 56

If $a=4, b=5$ and $\cos (A-B)=\frac{31}{32}$, prove that $c=6$.

Aman Gupta

Numerade Educator

03:17

Problem 57

If $a, b$ and $c$ be in A.P. prove that
i. $\cot \frac{A}{2}, \cot \frac{B}{2}$ and $\cot \frac{C}{2}$ are in A.P.
ii. $\cos A \cot \frac{A}{2}, \cos B \cot \frac{B}{2}$ and $\cos C \cot \frac{C}{2}$ are in A.P.
iii. $a \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{A}{2}=\frac{3 b}{2}$.
iv. $\tan \frac{A}{2}+\tan \frac{C}{2}=\frac{2}{3} \cot \frac{B}{2}$.
v. $\cot \frac{A}{2} \cot \frac{C}{2}=3$.

Aman Gupta

Numerade Educator

03:17

Problem 58

If $a^{2}, b^{2}$ and $c^{2}$ be in A.P., prove that
i. $\cot A, \cot B$ and $\cot C$ are in A.P.
ii. $\frac{\sin 3 B}{\sin B}=\left[\frac{a^{2}-c^{2}}{2 a c}\right]^{2}$

Aman Gupta

Numerade Educator

02:53

Problem 59

If $a, b$ and $c$ are in H.P., prove that $\sin ^{2} \frac{A}{2}, \sin ^{2} \frac{B}{2}$ and $\sin ^{2} \frac{C}{2}$ are also in H.P.

Aman Gupta

Numerade Educator

03:04

Problem 60

The sides of a triangle are in A.P. and the greatest and least angles are $\theta$ and $\phi$; prove that $4(1-\cos \theta)(1-\cos \phi)=\cos \theta+\cos \phi$

Aman Gupta

Numerade Educator

04:42

Problem 61

The sides of a triangle are in A.P. and the greatest angle exceeds the least by $90^{\circ}$; prove that the sides are proportional to $\sqrt{7}+1, \sqrt{7}$ and $\sqrt{7}-1$.

Aman Gupta

Numerade Educator

01:25

Problem 62

If $C=60^{\circ}$, then prove that $\frac{1}{a+c}+\frac{1}{b+c}=\frac{3}{a+b+c}$.

Aman Gupta

Numerade Educator

01:20

Problem 63

In any triangle prove that, if $\theta$ be any angle, then $b \cos \theta=c \cos (A-\theta)+a \cos (C+\theta)$.

Aman Gupta

Numerade Educator

02:00

Problem 64

If $A=45^{\circ}, B=75^{\circ}$, prove that $a+c \sqrt{2}=2 b$.

Aman Gupta

Numerade Educator

01:41

Problem 65

If $(a+b+c)(b+c-a)=k b c$, then prove that $k \in(0,4)$.

Aman Gupta

Numerade Educator

Problem 66

The sides of a triangle are $a, b, \sqrt{a^{2}+a b+b^{2}}$, prove that the greatest angle is $120^{\circ} .$

Aman Gupta

Numerade Educator

Problem 67

In any triangle $A B C$, if $\sin ^{2} A+\sin ^{2} B=\sin ^{2} C$, then show that the triangle is right angled.

Aman Gupta

Numerade Educator

01:09

Problem 68

In any $\triangle A B C$ if $2 \cos B=\frac{a}{c}$, then show that the triangle is isosceles.

Aman Gupta

Numerade Educator

Problem 69

If in a $\triangle A B C, a \sin A=b \sin B$, then show that the triangle is isosceles.

Aman Gupta

Numerade Educator

01:08

Problem 70

If $\cos A=\frac{\sin B}{2 \sin C}$, prove that the triangle is isosceles.

Aman Gupta

Numerade Educator

04:22

Problem 71

If $\frac{\cos A+2 \cos C}{\cos A+2 \cos B}=\frac{\sin B}{\sin C}$, prove that the triangle is either isosceles or right angled.

Aman Gupta

Numerade Educator

02:25

Problem 72

If $\frac{2 \cos A}{a}+\frac{\cos B}{b}+\frac{2 \cos C}{c}=\frac{a}{b c}+\frac{b}{c a}$, find the value of $A .\left\{\right.$ Ans. $\left.90^{\circ}\right\}$

Aman Gupta

Numerade Educator

02:38

Problem 73

If $A, B, C$ are in A.P., show that $2 \cos \frac{A-C}{2}=\frac{a+c}{\sqrt{a^{2}-a c+c^{2}}}$.

Aman Gupta

Numerade Educator

Problem 74

If the angles of a triangle are in the ratio $1: 2: 4$, then prove that $a^{2} b^{2} c^{2}=\left(b^{2}-a^{2}\right)\left(c^{2}-b^{2}\right)\left(c^{2}-a^{2}\right)$.

Aman Gupta

Numerade Educator

Problem 75

If $a \cos A=b \cos B$, prove that the triangle is either isosceles or right angled.

Aman Gupta

Numerade Educator

03:09

Problem 76

$\frac{a^{2}-b^{2}}{a^{2}+b^{2}}=\frac{\sin (A-B)}{\sin (A+B)}$, prove that the triangle is either isosceles or right angled.

Aman Gupta

Numerade Educator

02:29

Problem 77

If $\cos ^{2} A+\cos ^{2} B+\cos ^{2} C=1$, prove that the triangle is right angled.

Aman Gupta

Numerade Educator

Problem 78

If $\cot A+\cot B+\cot C=\sqrt{3}$, prove that the triangle is equilateral.

Aman Gupta

Numerade Educator

03:11

Problem 79

If $\cos A \cos B+\sin A \sin B \sin C=1$, show that $a: b: c=1: 1: \sqrt{2}$.

Aman Gupta

Numerade Educator

02:58

Problem 80

If $\cos A+2 \cos B+\cos C=2$, prove that the sides of the triangle are in A.P.

Aman Gupta

Numerade Educator

01:38

Problem 81

$\frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}$, prove that $a^{2}, b^{2}, c^{2}$ are in A.P.

Aman Gupta

Numerade Educator

Problem 82

If $b+c=3 a$, prove that $\cot \frac{B}{2} \cot \frac{C}{2}=2$.

Aman Gupta

Numerade Educator

03:06

Problem 83

In a triangle $A B C, \angle B=\frac{\pi}{3}$ and $\angle C=\frac{\pi}{4} .$ Let $D$ divide $B C$ internally in the ratio $1: 3$, then find the value of $\frac{\sin \angle B A D}{\sin \angle C A D}$.

Aman Gupta

Numerade Educator

02:45

Problem 84

The sides of a triangle are three consecutive natural numbers and it's largest angle is twice the smallest one. Determine the sides of the triangle.

Aman Gupta

Numerade Educator

02:46

Problem 85

Find the greatest angle of the triangle whose sides are $x^{2}+x+1,2 x+1$ and $x^{2}-1$.

Aman Gupta

Numerade Educator

03:41

Problem 86

If $a \tan A+b \tan B=(a+b) \tan \left(\frac{A+B}{2}\right)$, prove that $A=B$.

Aman Gupta

Numerade Educator

02:54

Problem 87

If $c(a+b) \cos \frac{B}{2}=b(a+c) \cos \frac{C}{2}$, prove that the triangle is isosceles.

Aman Gupta

Numerade Educator

02:08

Problem 88

If $B=3 C$, prove that $\cos C=\sqrt{\frac{b+c}{4 c}}, \sin C=\sqrt{\frac{3 c-b}{4 c}}$ and $\sin \frac{A}{2}=\frac{b-c}{2 c}$.

Aman Gupta

Numerade Educator

02:52

Problem 89

If $\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}$, then prove that $\frac{\cos A}{7}=\frac{\cos B}{19}=\frac{\cos C}{25}$.

Aman Gupta

Numerade Educator

03:43

Problem 90

If $a, b, c$ be $5,4,3$ respectively and $D$ and $E$ are the points of trisection of side $B C$, then prove that $\tan \angle C A E=\frac{3}{8}$.

Aman Gupta

Numerade Educator

03:21

Problem 91

$A B C D$ is a trapezium such that $A B$ is parallel to $C D$ and $C B$ is perpendicular to them. If $\angle A D B=\theta, B C=$ $p$ and $C D=q$, show that $A B=\frac{\left(p^{2}+q^{2}\right) \sin \theta}{p \cos \theta+q \sin \theta}$

Aman Gupta

Numerade Educator

03:19

Problem 92

If the tangents of the angles of a triangle are in A.P., prove that the squares of the sides are in the ratio $x^{2}\left(x^{2}+9\right):\left(3+x^{2}\right)^{2}: 9\left(1+x^{2}\right)$, where $x$ is the tangent of the least or greatest angle.

Aman Gupta

Numerade Educator

03:29

Problem 93

If $p$ and $q$ be perpendiculars from the angular points $A$ and $B$ on any line passing through the vertex $C$ of the triangle $A B C$, then prove that $a^{2} p^{2}+b^{2} q^{2}-2 a b p q \cos C=a^{2} b^{2} \sin ^{2} C$.

Aman Gupta

Numerade Educator

05:34

Problem 94

In the triangle $A B C$, lines $O A, O B$ and $O C$ are drawn so that the angles $O A B, O B C$ and $O C A$ are each equal to $\omega$; prove that $\cot \omega=\cot A+\cot B+\cot C$ and $\operatorname{cosec}^{2} \omega=\operatorname{cosec}^{2} A+\operatorname{cosec}^{2} B+\operatorname{cosec}^{2} C$

Aman Gupta

Numerade Educator

05:34

Problem 95

In any triangle $A B C$ if $D$ be any point of the base $B C$, such that $B D: D C=m: n$, and if $\angle B A D=\alpha, \angle D A C$ $=\beta$, and $\angle C D A=\theta$ and $A D=x$, prove that $(m+n) \cot \theta=m \cot \alpha-n \cot \beta=n \cot B-m \cot C$ and
$(m+n)^{2} \cdot x^{2}=(m+n)\left(m b^{2}+n c^{2}\right)-m n a^{2}$

Aman Gupta

Numerade Educator

02:07

Problem 96

Two straight roads intersect at an angle of $60^{\circ} .$ A bus on one road is $2 \mathrm{~km}$. away from the intersection and a car on the other is $3 \mathrm{~km}$. away from the intersection. Find the direct distance between the two vehicles

Urvashi Arora

Numerade Educator

Problem 97

A ring, $10 \mathrm{~cm}$. in diameter, is suspended from a point $12 \mathrm{~cm}$. above its center by 6 equal strings attached to its circumference at equal intervals. Find the cosine of the angle between consecutive strings.

Aman Gupta

Numerade Educator

03:36

Problem 98

The side of a base of a square pyramid is $a$ meters and it's vertex is at a height of $h$ meters above the center of the base. If $\theta \& \phi$ be respectively the inclinations of any face to the base and of any two faces to one another, prove that $\tan \theta=\frac{2 h}{a}$ and $\cot \frac{\phi}{2}=\sqrt{1+\frac{a^{2}}{2 h^{2}}}$.

Aman Gupta

Numerade Educator

01:30

Problem 99

Find the area of the triangle having sides 13,14 and $15 \mathrm{~cm}$.

Aman Gupta

Numerade Educator

01:39

Problem 100

If the angles of a triangle are $30^{\circ}$ and $45^{\circ}$ and the included side is $(\sqrt{3}+1) \mathrm{cm} .$, then prove that the area of the triangle is $\frac{1}{2}(\sqrt{3}+1)$ sq. $\mathrm{cm}$.

Aman Gupta

Numerade Educator

Problem 101

If $B=45^{\circ}, a=2(\sqrt{3}+1)$ units and $\Delta=6+2 \sqrt{3}$ sq. units. Determine the side $b$.

Aman Gupta

Numerade Educator

03:03

Problem 102

If one angle of a triangle be $60^{\circ}$, the area $10 \sqrt{3}$ sq. $\mathrm{cm}$. and the perimeter $20 \mathrm{~cm} .$, find the length of the sides.

Aman Gupta

Numerade Educator

01:45

Problem 103

If $a=6, b=3$ and $\cos (A-B)=\frac{4}{5}$, then find it's area.

Aman Gupta

Numerade Educator

01:54

Problem 104

If $\Delta=a^{2}-(b-c)^{2}$, find $\tan \frac{A}{2}$

Aman Gupta

Numerade Educator

05:10

Problem 105

If $C=60^{\circ}, A=75^{\circ}$ and $D$ is a point on $A C$ such that the area of the $\Delta B A D$ is $\sqrt{3}$ times the area of the $\Delta B C D$, find the $\angle A B D .\left\{\right.

Aman Gupta

Numerade Educator

05:51

Problem 106

If the sides of a triangle are in A.P. and it's area is $\frac{3}{e}$ th of an equilateral triangle of same perimeter. Prove 8
that it's sides are in the ratio $3: 5: 7$.

Aman Gupta

Numerade Educator

Problem 107

If $p_{1}, p_{2}, p_{3}$ are altitudes of a triangle $A B C$ from the vertices $A, B, C$ and $\Delta$ the area of the triangle, then prove that $p_{1}^{-2}+p_{2}^{-2}+p_{3}^{-2}=\frac{a^{2}+b^{2}+c^{2}}{4 \Delta^{2}}$.

Aman Gupta

Numerade Educator

04:32

Problem 108

If $\frac{a}{1+m^{2} n^{2}}=\frac{b}{m^{2}+n^{2}}=\frac{c}{\left(1-m^{2}\right)\left(1+n^{2}\right)}$, then prove that $\tan \frac{A}{2}=\frac{m}{n}, \tan \frac{B}{2}=m n$ and $\Delta=\frac{m n b c}{m^{2}+n^{2}}$.

Aman Gupta

Numerade Educator

03:43

Problem 109

The sides $a, b, c$ of a triangle are the roots of the equation $x^{3}-p x^{2}+q x-r=0$. Prove that $\Delta=\frac{1}{4} \sqrt{p\left(4 p q-p^{3}-8 r\right)} .$

Aman Gupta

Numerade Educator

02:53

Problem 110

Through the angular points of a triangle are drawn straight lines which make the same angle $\alpha$ with the opposite sides of the triangle. Prove that the area of the triangle formed by them is to the area of the original triangle as $4 \cos ^{2} \alpha: 1$.

Teresa Fuston

Numerade Educator

Problem 111

In the sides $B C, C A$ and $A B$ are taken three points $A^{\prime}, B^{\prime}, C^{\prime}$ such that $B A^{\prime}: A^{\prime} C=C B^{\prime}: B^{\prime} A=A C^{\prime}: C^{\prime} B=$
$m: n$. Prove that if $A A^{\prime}, B B^{\prime}$ and $C C^{\prime}$ be joined, they will form by their intersections a triangle whose area is to that of the triangle $A B C$ as $(m-n)^{2}: m^{2}+m n+n^{2}$

Aman Gupta

Numerade Educator

01:10

Problem 112

In a triangle $A B C, a=4, b=3, \angle A=60^{\circ}$. Then show that $c$ is the root of the equation $c^{2}-3 c-7=0$.

Aman Gupta

Numerade Educator

03:41

Problem 113

In the ambiguous case, given $a, b, A$ and $c_{1}, c_{2}$ are the two value of side $c$, then show that i. $\quad c_{1}+c_{2}=2 b \cos A$.
ii. $\quad c_{1} c_{2}=b^{2}-a^{2}$.
iii. $c_{1} \sim c_{2}=2 \sqrt{a^{2}-b^{2} \sin ^{2} A}$.
iv. $c_{1}^{2}-2 c_{1} c_{2} \cos 2 A+c_{2}^{2}=4 a^{2} \cos ^{2} A$.
v. $\left(c_{1}-c_{2}\right)^{2}+\left(c_{1}+c_{2}\right)^{2} \tan ^{2} A=4 a^{2}$.
vi. $\frac{(a+b)^{2}}{1+\cos C_{1}}+\frac{(b-a)^{2}}{1-\cos C_{1}}=\frac{(a+b)^{2}}{1+\cos C_{2}}+\frac{(b-a)^{2}}{1-\cos C_{2}}$.
vii. $\cos B_{1} C B_{2}=\frac{2 c_{1} c_{2}}{c_{1}^{2}+c_{2}^{2}}$ if $A=45^{\circ}$.

Aman Gupta

Numerade Educator

03:41

Problem 114

In the ambiguous case, given $a, c, A$ and $b_{2}=2 b_{1}$, where $b_{1}, b_{2}$ are the two value of side $b$, then prove that $3 a=c \sqrt{1+8 \sin ^{2} A}$

Aman Gupta

Numerade Educator

04:52

Problem 115

In the ambiguous case, given $a, b, A$, if the remaining angles of the triangles formed be $B_{1}, C_{1}$ and $B_{2}$, $C_{2}$, then prove that $\frac{\sin C_{1}}{\sin B_{1}}+\frac{\sin C_{2}}{\sin B_{2}}=2 \cos A$.

Aman Gupta

Numerade Educator

03:09

Problem 116

In the ambiguous case, given $a, b, A$, prove that the sum of the areas of the two triangles formed is $\frac{1}{2} b^{2} \sin 2$

Aman Gupta

Numerade Educator

05:18

Problem 117

If $2 b=3 a$ and $\tan ^{2} A=\frac{3}{5}$, prove that there are two values of $c$, one of which is double the other.

Aman Gupta

Numerade Educator

03:27

Problem 118

If $2 b=(m+1) a$ and $\cos A=\frac{1}{2} \sqrt{\frac{(m-1)(m+3)}{m}}$, where $1<m<3$, then prove that there are two values of the third side, one of which is $m$ times the other.

Aman Gupta

Numerade Educator

02:56

Problem 119

Determine the lengths of medians in terms of the sides.

Charles Carter

Numerade Educator

Problem 120

If $D$ is the mid-point of $B C$, then prove that $\sin \angle C A D=\frac{a \sin C}{\sqrt{2 b^{2}+2 c^{2}-a^{2}}}$, $\sin \angle B A D=\frac{a \sin B}{\sqrt{2 b^{2}+2 c^{2}-a^{2}}}, \sin \angle A D B=\frac{2 b \sin C}{\sqrt{2 b^{2}+2 c^{2}-a^{2}}}$ and $\cot \angle A D B=\frac{\left(b^{2}-c^{2}\right)}{4 \Delta} .$

Aman Gupta

Numerade Educator

Problem 121

In a triangle $A B C$, the median to the side $B C$ is of length $\frac{1}{\sqrt{11-6 \sqrt{3}}}$ and it divides angle $A$ into angles of $30^{\circ}$ and $45^{\circ}$. Prove that side $B C$ is of length 2 units.

Anurag Kumar

Numerade Educator

01:48

Problem 122

In an isosceles right-angled triangle a straight line is drawn from the middle point of one of the equal sides to the opposite angle. Show that it divides the angle into parts whose cotangents are 2 and 3 .

Aman Gupta

Numerade Educator

02:13

Problem 123

If in a triangle the median through $A$ is perpendicular to the side $A B$, prove that $\tan A+2 \tan B=0$.

Aman Gupta

Numerade Educator

03:37

Problem 124

If $D$ is the mid-point of $B C$ and $A D$ is perpendicular to $A C$, then prove that $\cos A \cos C=\frac{2\left(c^{2}-a^{2}\right)}{3 a c}$.

Aman Gupta

Numerade Educator

04:43

Problem 125

Prove that the median through $A$ divides it into angles whose cotangents are $2 \cot A+\cot C$ and $2 \cot A+\cot B$ and makes with the base an angle whose cotangent is $\frac{1}{2}(\cot C \sim \cot B)$.

Aman Gupta

Numerade Educator

Problem 126

The sides of a right angled triangle are 21 and $28 \mathrm{~cm}$.; find the length of the perpendicular drawn to the hypotenuse from the right angle.

Aman Gupta

Numerade Educator

02:21

Problem 127

The perpendicular $A D$ to the base of a triangle $A B C$ divides it into segments such that $B D, C D$ and $A D$ are in the ratio of 2,3 and $6 ;$ prove that the vertical angle of the triangle is $45^{\circ}$.

Aman Gupta

Numerade Educator

02:18

Problem 128

If $\sin A, \sin B, \sin C$ are in A.P., then prove that the altitudes are in H.P.

Aman Gupta

Numerade Educator

03:12

Problem 129

If $A D$ is the altitude from $A, b>c, C=23^{\circ}$ and $A D=\frac{a b c}{b^{2}-c^{2}}$, find $B$.

Aman Gupta

Numerade Educator

01:41

Problem 130

The base angles of a triangle are $22 \frac{1}{2}^{\circ}$ and $112 \frac{1}{2} \circ$. Show that the base is equal to twice the height.

Aman Gupta

Numerade Educator

Problem 131

Prove that the perpendicular from $A$ divides $B C$ into portions which are proportional to the cotangents of the adjacent angles and that it divides the angle $A$ into portions whose cosines are inversely proportional to the adjacent sides.

Aman Gupta

Numerade Educator

01:20

Problem 132

Prove that the distance between the middle point of $B C$ and the foot of the perpendicular from $A$ is $\frac{b^{2} \sim c^{2}}{2 a}$

Aman Gupta

Numerade Educator

02:07

Problem 133

If $p, q, r$ are the altitudes of a triangle $A B C$, prove that $\frac{1}{p^{2}}+\frac{1}{q^{2}}+\frac{1}{r^{2}}=\frac{\cot A+\cot B+\cot C}{\Delta}$.

Aman Gupta

Numerade Educator

Problem 134

If $p_{1}, p_{2}, p_{3}$ are altitudes of a triangle $A B C$ from the vertices $A, B, C$ and $\Delta$ the area of the triangle,
then prove that $p_{1}^{-1}+p_{2}^{-1}-p_{3}^{-1}=\frac{s-c}{\Delta}$.

Aman Gupta

Numerade Educator

01:08

Problem 135

If $p, q, r$ are the altitudes of a triangle from the vertices $A, B, C$ respectively, prove that $\frac{1}{p}+\frac{1}{q}-\frac{1}{r}=\frac{a b}{s \Delta} \cos ^{2} \frac{C}{2}$

Aman Gupta

Numerade Educator

02:02

Problem 136

If $x, y, z$ are respectively the distance of the vertices from orthocentre, prove that $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{a b c}{x y z}$.

Aman Gupta

Numerade Educator

04:28

Problem 137

In a triangle of base $a$, the ratio of the other sides is $r(r<1)$. Show that the altitude of the triangle is less than or equal to $\frac{a r}{1-r^{2}}$.

Aman Gupta

Numerade Educator

01:45

Problem 138

Prove that the length of internal bisector $A D$ is $\frac{2 b c}{b+c} \cos \frac{A}{2}$.

Aman Gupta

Numerade Educator

01:42

Problem 139

If $p, q, r$ are the lengths of internal bisectors of the angles $A, B, C$ respectively, then prove that $\frac{1}{p} \cos \frac{A}{2}+\frac{1}{q} \cos \frac{B}{2}+\frac{1}{r} \cos \frac{C}{2}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Aman Gupta

Numerade Educator

01:26

Problem 140

In a right angled triangle $A B C$, the bisector of the right angle $C$ divides $A B$ into segments $p$ and $q$ and if $\tan \frac{A-B}{2}=t$, then show that $p: q=(1-t):(1+t)$.

Aman Gupta

Numerade Educator

Problem 141

If the bisectors of the angles of a triangle $A B C$ meet the opposite sides in $A^{\prime}, B^{\prime}$ and $C^{\prime}$, prove that the ratio of the areas of the triangles $A^{\prime} B^{\prime} C^{\prime}$ and $A B C$ is $2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}: \cos \frac{A-B}{2} \cos \frac{B-C}{2} \cos \frac{C-A}{2}$

Aman Gupta

Numerade Educator

Problem 142

$\Delta=2 R^{2} \sin A \sin B \sin C$

Aman Gupta

Numerade Educator

05:15

Problem 143

$4 R \sin A \sin B \sin C=a \cos A+b \cos B+c \cos C$

Aman Gupta

Numerade Educator

Problem 144

$\sin A+\sin B+\sin C=\frac{s}{R}$

Aman Gupta

Numerade Educator

01:34

Problem 145

The sides of a triangle are 18,24 and $30 \mathrm{~cm}$. Find $R$.

Aman Gupta

Numerade Educator

01:14

Problem 146

Find the circumradius of the equilateral triangle of side $2 \sqrt{3} \mathrm{~cm}$.

Aman Gupta

Numerade Educator

02:18

Problem 147

In the ambiguous case of the triangle, prove that the circumradius of the two triangles are equal.

Suman Saurav Thakur

Numerade Educator

06:09

Problem 148

Prove that
i. $O P^{2}=9 R^{2}-\left(a^{2}+b^{2}+c^{2}\right)$.
ii. $O G^{2}=R^{2}-\frac{1}{9}\left(a^{2}+b^{2}+c^{2}\right)$.

Suman Saurav Thakur

Numerade Educator

Problem 149

If $x, y, z$ are respectively the perpendiculars from the vertices $A, B, C$ to the opposite sides, prove that
i. $x y z=\frac{a^{2} b^{2} c^{2}}{8 R^{3}}$
ii. $\frac{\cos A}{x}+\frac{\cos B}{y}+\frac{\cos C}{z}=\frac{1}{R}$
iii. $\frac{b x}{c}+\frac{c y}{a}+\frac{a z}{b}=\frac{a^{2}+b^{2}+c^{2}}{2 R}$

Aman Gupta

Numerade Educator

Problem 150

If $x, y, z$ are respectively the perpendiculars from the circumcentre to the sides, prove that
$\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{a b c}{4 x y z}$

Aman Gupta

Numerade Educator

02:29

Problem 151

If $8 R^{2}=a^{2}+b^{2}+c^{2}$, prove that the triangle is right angled.

Aman Gupta

Numerade Educator

Problem 152

Prove that the radii of the circles circumscribing the triangles $B P C, C P A, A P B$ and $A B C$ are all equal.

Wen Zheng

Numerade Educator

02:35

Problem 153

$D, E$ and $F$ are the middle points of the sides of the triangle $A B C$; prove that the centroid of the triangle $D E F$ is the same as that of $A B C$ and that it's orthocentre is the circumcentre of $A B C .$

R M

Numerade Educator

07:43

Problem 154

If $R_{1}, R_{2}$ and $R_{3}$ are respectively the radii of the circumcircles of the triangle $O B C, O C A$ and $O A B$, prove that $\frac{a}{R_{1}}+\frac{b}{R_{2}}+\frac{c}{R_{3}}=\frac{a b c}{R^{3}}$.

Gaurav Kalra

Numerade Educator

Problem 155

At the points $A, B, C$, tangents are drawn to the circumcircle. These tangents enclose a triangle $P Q R$. Prove that its angles and sides are respectively $180^{\circ}-2 A, 180^{\circ}-2 B, 180^{\circ}-2 C$ and $\frac{a}{2 \cos B \cos C}$,
$\frac{b}{2 \cos C \cos A}, \frac{c}{2 \cos A \cos B}$

Aman Gupta

Numerade Educator

04:55

Problem 156

The legs of a tripod are each $10 \mathrm{~cm}$. in length and their points of contact with a horizontal table on which the tripod stands form a triangle whose sides are 7,8 and $9 \mathrm{~cm}$. in length. Find the inclination of the legs to the horizontal and the height of the apex.

Khoobchandra Agrawal

Numerade Educator

01:40

Problem 157

$$
\Delta=4 R r \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2} \text { . }
$$

Aman Gupta

Numerade Educator

Problem 158

$$
\frac{1}{b c}+\frac{1}{c a}+\frac{1}{a b}=\frac{1}{2 R r}
$$

Aman Gupta

Numerade Educator

01:19

Problem 159

$\cos A+\cos B+\cos C=1+\frac{r}{R}$

Aman Gupta

Numerade Educator

01:30

Problem 160

$a \cot A+b \cot B+c \cot C=2(R+r)$

Aman Gupta

Numerade Educator

01:58

Problem 161

$(b+c) \tan \frac{A}{2}+(c+a) \tan \frac{B}{2}+(a+b) \tan \frac{C}{2}=4(R+r)$

Aman Gupta

Numerade Educator

01:43

Problem 162

$\cos ^{2} \frac{A}{2}+\cos ^{2} \frac{B}{2}+\cos ^{2} \frac{C}{2}=2+\frac{r}{2 R}$

Aman Gupta

Numerade Educator

Problem 163

Find the in-radius of the triangle having sides $13,14,15$.

Aman Gupta

Numerade Educator

01:30

Problem 164

The sides of a triangle are 18,24 and $30 \mathrm{~cm}$. Find $r$.

Aman Gupta

Numerade Educator

02:16

Problem 165

Given $a: b: c=4: 5: 6$. Find the ratio of the radius of the circumcircle to that of the incircle. \{ns. $\left.\frac{16}{7}\right\}$

Aman Gupta

Numerade Educator

02:22

Problem 166

If $C=90^{\circ}$, prove that $R+r=\frac{1}{2}(a+b)$ and $\frac{c}{r}=\frac{c+a}{b}+\frac{c+b}{a}$.

Aman Gupta

Numerade Educator

01:39

Problem 167

Prove that
i. $\quad A I=r \operatorname{cosec} \frac{A}{2}$.
ii. $\quad I A \cdot I B \cdot I C=a b c \tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}=4 R r^{2}$.
iii. $I O^{2}=R^{2}(3-2 \cos A-2 \cos B-2 \cos C)$.
iv. $I P^{2}=2 r^{2}-4 R^{2} \cos A \cos B \cos C$.
v. Area of $\Delta I O P=2 R^{2} \sin \frac{B-C}{2} \sin \frac{C-A}{2} \sin \frac{A-B}{2}$.
vi. Area of $\Delta I P G=\frac{4}{3} R^{2} \sin \frac{B-C}{2} \sin \frac{C-A}{2} \sin \frac{A-B}{2}$.

Rukhmani Jain

Numerade Educator

05:35

Problem 168

$D E F$ is the triangle formed by joining the points of contact of the incircle with the sides of the triangle $A B C$; prove that:-
i. it's sides are $2 r \cos \frac{A}{2}, 2 r \cos \frac{B}{2}, 2 r \cos \frac{C}{2}$
ii. it's angles are $\frac{\pi}{2}-\frac{A}{2}, \frac{\pi}{2}-\frac{B}{2}, \frac{\pi}{2}-\frac{C}{2}$
iii. it's area is $\frac{2 \Delta^{3}}{a b c s}$, i.e. $\frac{r \Delta}{2 R}$.

Sandip Ranjan

Numerade Educator

Problem 169

$A D, B E$ and $C F$ are the perpendiculars from the angular points of a triangle $A B C$ upon the opposite sides. Prove that the diameters of the circumcircles of the triangles $A E F, B D F$ and $C D E$ are respectively $a \cot A$, $b \cot B$ and $c \cot C$, and that the perimeters of the triangles $D E F$ and $A B C$ are in the ratio $r: R$.

Wen Zheng

Numerade Educator

Problem 170

If $x, y, z$ are respectively the perpendiculars from the vertices $A, B, C$ to the opposite sides, prove that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{r}$

Aman Gupta

Numerade Educator

02:02

Problem 171

If $x, y, z$ are respectively the distance of the vertices from orthocentre, prove that $x+y+z=2(R+r)$.

Aman Gupta

Numerade Educator

01:38

Problem 172

If $x, y, z$ are the distances of the vertices of a triangle from the corresponding points of contact with the incircle, prove that $\frac{x y z}{(x+y+z)}=r^{2}$.

Aman Gupta

Numerade Educator

Problem 173

In a $\Delta A B C$, the line joining the circumcentre to the incentre is parallel to $B C$, then find the value of $\cos B+\cos C .\{$ Ans. 1$\}$

Aman Gupta

Numerade Educator

06:31

Problem 174

Let $A B C$ be a triangle and let $B B_{1}, C C_{1}$ be respectively the bisectors of $\angle B, \angle C$ with $B_{1}$ on $A C$ and $C_{1}$ on $A B$. Let $E, F$ be the feet of perpendiculars drawn from $A$ on $B B_{1}, C C_{1}$ respectively. Suppose $D$ is the point at which the incircle of $A B C$ touches $A B$. Prove that $A D=E F$.

Nidhi Garg

Numerade Educator

05:35

Problem 175

Let $A B C$ be a triangle having $O$ and $I$ as its circumcentre and incentre respectively. If $R$ and $r$ are the circumradius and the inradius respectively, then prove that $(I O)^{2}=R^{2}-2 R r$. Further show that the triangle $B I O$ is right-angled triangle if and only if $b$ is the arithmetic mean of $a$ and $c$.

Sandip Ranjan

Numerade Educator

Problem 176

If $R_{1}, R_{2}$ and $R_{3}$ are respectively the radii of the circumcircles of the triangle $I B C, I C A$ and $I A B$, prove that $R_{1} R_{2} R_{3}=2 R^{2} r$.

Aman Gupta

Numerade Educator

01:31

Problem 177

If $x, y, z$ be the lengths of the tangents drawn to the incircle parallel to the sides $B C, A C, A B$ respectively and intercepted between the other two sides, then prove that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$.

Dr Harish Viswanathan

Numerade Educator

02:50

Problem 178

Prove that the area of the incircle is to the area of the triangle itself is $\pi: \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}$.

Aman Gupta

Numerade Educator

01:53

Problem 179

$C$ is right-angled and a perpendicular $C D$ is drawn to $A B$. The radii of the circles inscribed into the triangles $A C D$ and $B C D$ are equal to $x$ and $y$ respectively. Find the radius of the circle inscribed into the triangle $A B C$.

Aman Gupta

Numerade Educator

02:31

Problem 180

If a circle be drawn touching the incircle and circumcircle of a triangle and the side $B C$ externally, prove that it's radius is $\frac{\Delta}{a} \tan ^{2} \frac{A}{2}$.

Jay Patel

Numerade Educator

01:06

Problem 181

$\frac{r r_{1}}{r_{2} r_{3}}=\tan ^{2} \frac{A}{2}$

Aman Gupta

Numerade Educator

Problem 182

$$
r r_{1} r_{2} r_{3}=\Delta^{2}
$$

Aman Gupta

Numerade Educator

02:50

Problem 183

$r_{1} r_{2} r_{3}=r^{3} \cot ^{2} \frac{A}{2} \cot ^{2} \frac{B}{2} \cot ^{2} \frac{C}{2}$

Aman Gupta

Numerade Educator

01:08

Problem 184

PROVING IDENTITIES RELATED TO EX-RADII
$$
r r_{1} \cot \frac{A}{2}=\Delta
$$

Aman Gupta

Numerade Educator

01:32

Problem 185

PROVING IDENTITIES RELATED TO EX-RADII
$$
r_{2} r_{3}+r_{3} r_{1}^{*}+r_{1} r_{2}=s^{2}
$$

Aman Gupta

Numerade Educator

Problem 186

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}=\frac{1}{r}
$$

Aman Gupta

Numerade Educator

03:17

Problem 187

PROVING IDENTITIES RELATED TO EX-RADII
$$
a\left(r r_{1}+r_{2} r_{3}\right)=b\left(r r_{2}+r_{3} r_{1}\right)=c\left(r r_{3}+r_{1} r_{2}\right)
$$

Aman Gupta

Numerade Educator

02:38

Problem 188

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(r_{1}+r_{2}\right) \tan \frac{C}{2}=\left(r_{3}-r\right) \cot \frac{C}{2}=c
$$

Aman Gupta

Numerade Educator

01:40

Problem 189

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{1}{r^{2}}+\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}+\frac{1}{r_{3}^{2}}=\frac{a^{2}+b^{2}+c^{2}}{\Delta^{2}}
$$

Aman Gupta

Numerade Educator

02:43

Problem 190

PROVING IDENTITIES RELATED TO EX-RADII
$$
r_{1}+r_{2}+r_{3}-r=4 R
$$

Aman Gupta

Numerade Educator

02:04

Problem 191

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(r_{1}-r\right)\left(r_{2}-r\right)\left(r_{3}-r\right)=4 R r^{2}
$$

Aman Gupta

Numerade Educator

02:26

Problem 192

$$
\frac{r_{1}}{b c}+\frac{r_{2}}{c a}+\frac{r_{3}}{a b}=\frac{1}{r}-\frac{1}{2 R}
$$

Aman Gupta

Numerade Educator

02:53

Problem 193

PROVING IDENTITIES RELATED TO EX-RADII
$$
r^{2}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=16 R^{2}-a^{2}-b^{2}-c^{2}
$$

Aman Gupta

Numerade Educator

Problem 194

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{\left(r_{1}-r\right)}{a}+\frac{\left(r_{2}-r\right)}{b}=\frac{c}{r_{3}}
$$

Aman Gupta

Numerade Educator

03:01

Problem 195

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{r_{1}\left(r_{2}+r_{3}\right)}{a}=\frac{r_{2}\left(r_{3}+r_{1}\right)}{b}=\frac{r_{3}\left(r_{1}+r_{2}\right)}{c}
$$

Aman Gupta

Numerade Educator

02:30

Problem 196

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{\left(a b-r_{1} r_{2}\right)}{r_{3}}=\frac{\left(b c-r_{2} r_{3}\right)}{r_{1}}=\frac{\left(c a-r_{3} r_{1}\right)}{r_{2}}=r
$$

Aman Gupta

Numerade Educator

Problem 197

PROVING IDENTITIES RELATED TO EX-RADII
$$
r_{1}+r_{2}-r_{3}+r=4 R \cos C
$$

Aman Gupta

Numerade Educator

Problem 198

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(r_{1}-r\right)\left(r_{2}+r_{3}\right)=a^{2}
$$

Aman Gupta

Numerade Educator

02:45

Problem 199

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(\frac{1}{r}-\frac{1}{r_{1}}\right)\left(\frac{1}{r}-\frac{1}{r_{2}}\right)\left(\frac{1}{r}-\frac{1}{r_{3}}\right)=\frac{16 R}{r^{2}(a+b+c)^{2}}
$$

Aman Gupta

Numerade Educator

Problem 200

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\left(\frac{1}{r_{2}}+\frac{1}{r_{3}}\right)\left(\frac{1}{r_{3}}+\frac{1}{r_{1}}\right)=\frac{64 R^{3}}{a^{2} b^{2} c^{2}}
$$

Aman Gupta

Numerade Educator

01:37

Problem 201

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(r_{1}+r_{2}\right)\left(r_{2}+r_{3}\right)\left(r_{3}+r_{1}\right)=4 R s^{2}
$$

Aman Gupta

Numerade Educator

Problem 202

PROVING IDENTITIES RELATED TO EX-RADII
$$
R=\frac{\left(r_{1}+r_{2}\right)\left(r_{2}+r_{3}\right)\left(r_{3}+r_{1}\right)}{4\left(r_{1} r_{2}+r_{2} r_{3}+r_{3} r_{1}\right)}
$$

Aman Gupta

Numerade Educator

01:04

Problem 203

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{r_{1}}{(s-b)(s-c)}+\frac{r_{2}}{(s-c)(s-a)}+\frac{r_{3}}{(s-a)(s-b)}=\frac{3}{r}
$$

Aman Gupta

Numerade Educator

02:21

Problem 204

PROVING IDENTITIES RELATED TO EX-RADII
$$
\left(r+r_{1}\right) \tan \frac{B-C}{2}+\left(r+r_{2}\right) \tan \frac{C-A}{2}+\left(r+r_{3}\right) \tan \frac{A-B}{2}=0
$$

Aman Gupta

Numerade Educator

01:26

Problem 205

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{(b-c)}{r_{1}}+\frac{(c-a)}{r_{2}}+\frac{(a-b)}{r_{3}}=0
$$

Aman Gupta

Numerade Educator

Problem 206

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{\left(r_{2}+r_{3}\right)}{(1+\cos A)}=\frac{\left(r_{3}+r_{1}\right)}{1+\cos B}=\frac{\left(r_{1}+r_{2}\right)}{1+\cos C}
$$

Aman Gupta

Numerade Educator

02:54

Problem 207

PROVING IDENTITIES RELATED TO EX-RADII
$$
\frac{b c}{r_{1}}+\frac{c a}{r_{2}}+\frac{a b}{r_{3}}=2 R\left[\left(\frac{a}{b}+\frac{b}{a}\right)+\left(\frac{b}{c}+\frac{c}{b}\right)+\left(\frac{c}{a}+\frac{a}{c}\right)-3\right]
$$

Aman Gupta

Numerade Educator

Problem 208

The sides of a triangle are 18,24 and $30 \mathrm{~cm}$. Find $r_{1}, r_{2}$ and $r_{3}$.

Aman Gupta

Numerade Educator

03:02

Problem 209

If $a=13, b=4$ and $\cos C=-\frac{5}{13}$, find $R, r, r_{1}, r_{2}$ and $r_{3} .$

Aman Gupta

Numerade Educator

Problem 210

If $r_{1}=2 r_{2}=3 r_{3}$ then prove that $a: b=5: 4$.

Aman Gupta

Numerade Educator

03:53

Problem 211

If $r_{1}, r_{2}, r_{3}$ are in H.P., area is 24 sq. $\mathrm{cm}$. and perimeter is $24 \mathrm{~cm} .$, find $a, b, c$. \{Ans. $\left.6 \mathrm{~cm} ., 8 \mathrm{~cm} ., 10 \mathrm{~cm} .\right\}$

Aman Gupta

Numerade Educator

Problem 212

Prove that
i. $\quad A I_{1}=r_{1} \operatorname{cosec} \frac{A}{2}$
ii. $I I_{1}=a \sec \frac{A}{2}$
iii. $I_{2} I_{3}=a \operatorname{cosec} \frac{A}{2}=4 R \cos \frac{A}{2}$
iv. $I I_{1} \cdot I I_{2} \cdot I I_{3}=16 R^{2} r$
v. $\quad I_{2} I_{3}{ }^{2}=4 R\left(r_{2}+r_{3}\right)$
vi. $\quad \angle I_{3} I_{1} I_{2}=\frac{B+C}{2}=\frac{\pi}{2}-\frac{A}{2}$
vii. $I I_{1}{ }^{2}+I_{2} I_{3}{ }^{2}=I I_{2}{ }^{2}+I_{3} I_{1}{ }^{2}=I I_{3}{ }^{2}+I_{1} I_{2}{ }^{2}$
viii. Area of $\Delta I_{1} I_{2} I_{3}=8 R^{2} \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}=\frac{a b c}{2 r}=2 R s$
ix. $\quad \frac{I I_{1} \cdot I_{2} I_{3}}{\sin A}=\frac{I I_{2} \cdot I_{3} I_{1}}{\sin B}=\frac{I I_{3} \cdot I_{1} I_{2}}{\sin C}$

Aman Gupta

Numerade Educator

Problem 213

Prove that the cubic equation whose roots are $r_{1}, r_{2}$ and $r_{3}$ is $x^{3}-(r+4 R) x^{2}+s^{2} x-s^{2} r=0$.

Hoan Nguyen

Numerade Educator

Problem 214

If $u, x, y, z$ are respectively the areas of the incircle and ex-circles, prove that $\frac{1}{\sqrt{u}}=\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}+\frac{1}{\sqrt{z}}$

Sikandar Baig

Numerade Educator

Problem 215

If $r_{1}=r_{2}+r_{3}+r$, prove that the triangle is right angled.

Aman Gupta

Numerade Educator

Problem 216

If $\left(1-\frac{r_{1}}{r_{2}}\right)\left(1-\frac{r_{1}}{r_{3}}\right)=2$, prove that the triangle is right angled.

Aman Gupta

Numerade Educator

07:07

Problem 217

If $(a-b)(s-c)=(b-c)(s-a)$, prove that $r_{1}, r_{2}, r_{3}$ are in A.P.

Sandip Ranjan

Numerade Educator

02:57

Problem 218

If $r_{1}, r_{2}, r_{3}$ are in H.P., show that $a, b, c$ are in A.P.

Sandip Ranjan

Numerade Educator

04:15

Problem 219

If $x, y, z$ be the lengths of the tangents from the centers of ex-circles to the circumcircle, prove that $\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{2 s}{a b c}$

Saurabh Chandra

Numerade Educator

Problem 220

If $\Delta_{0}$ be the area of the triangle formed by joining the points of contact of the incircle with the sides of the given triangle and $\Delta_{1}, \Delta_{2}$ and $\Delta_{3}$ the corresponding areas for the ex-circles, prove that $\Delta_{1}+\Delta_{2}+\Delta_{3}-\Delta_{0}=2 \Delta$

Aman Gupta

Numerade Educator

01:52

Problem 221

The triangle $D E F$ circumscribes the three ex-circles of the triangle $A B C$, prove that $\frac{E F}{a \cos A}=\frac{F D}{b \cos B}=\frac{D E}{c \cos C}$

Aaryan Kapoor

Numerade Educator

04:24

Problem 222

Two circles, of radii $a$ and $b$, cut each other at an angle $\theta$. Prove that the length of the common chord is $\frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}}$

Saurabh Chandra

Numerade Educator

00:30

Problem 223

Three equal circles touch one another. Find the radius of the circle which touches all three. \{Ans. $\left.\left(\frac{2}{\sqrt{3}} \pm 1\right) r\right\}$

James Kiss

Numerade Educator

01:14

Problem 224

Three circles, whose radii are $a, b$ and $c$, touch one another externally and the tangents at their points of contact meet in a point. Prove that the distance of this point from either of their points of contact is $\left(\frac{a b c}{a+b+c}\right)^{\frac{1}{2}}$

Sriparna Bhattacharjee

Numerade Educator

02:14

Problem 225

Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from any point of contact is 4 . Find the ratio of the product of the radii to the sum of the radii of circles.

Hast Aggarwal

Numerade Educator

Problem 226

In an acute angled triangle, prove that $\tan A+\tan B+\tan C \geq 3 \sqrt{3}$. If $\tan A+\tan B+\tan C=3 \sqrt{3}$, prove
that the triangle is equilateral.

Aman Gupta

Numerade Educator

Problem 227

Prove that $\cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} \geq 3 \sqrt{3}$

Aman Gupta

Numerade Educator

Problem 228

Prove that $1<\cos A+\cos B+\cos C \leq \frac{3}{2}$. In case of equality, triangle will be equilateral.

Aman Gupta

Numerade Educator

03:16

Problem 229

Prove that $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \leq \frac{1}{8}$.

Tanishq Gupta

Numerade Educator

02:06

Problem 230

Prove that $\Delta<\frac{s^{2}}{4}$

Linh Vu

Numerade Educator

04:16

Problem 231

The sides of a quadrilateral are respectively $3,4,5$ and $6 \mathrm{~cm} .$, and the sum of a pair of opposite angles is $120^{\circ}$. Find the area.

Teresa Fuston

Numerade Educator

02:08

Problem 232

Prove that the area of any quadrilateral is one-half the product of the two diagonals and the sine of the angle between them.

Ankit Gupta

Numerade Educator

00:40

Problem 233

$a, b, c$ and $d$ are the sides of a quadrilateral taken in order and $\alpha$ is the angle between the diagonals opposite to $b$ or $d$, prove that the area of the quadrilateral is $\frac{1}{4}\left(a^{2}-b^{2}+c^{2}-d^{2}\right) \tan \alpha$.

Nikhil Choudhary

Numerade Educator

Problem 234

If $a, b, c$ and $d$ be the sides and $x$ and $y$ the diagonals of a quadrilateral, prove that it's area is $\frac{1}{4} \sqrt{4 x^{2} y^{2}-\left(b^{2}+d^{2}-a^{2}-c^{2}\right)^{2}}$

Ashley High

Numerade Educator

01:25

Problem 235

The sides of a quadrilateral are divided in order in the ratio $m: n$ and a new quadrilateral is formed by joining the points of division. Prove that it's area is to the area of the original figure as $m^{2}+n^{2}$ to $(m+n)^{2} .$

Jay Patel

Numerade Educator

Problem 236

If $a, b, c$ and $d$ be the sides of a quadrilateral, taken in order, prove that $d^{2}=a^{2}+b^{2}+c^{2}-2 a b \cos B-2 b c \cos C-2 c a \cos \gamma$, where $\gamma$ denotes the angle between the sides $a$ and $c$.

Gio Maya

Numerade Educator

01:48

Problem 237

Sides of a cyclic quadrilateral $A B C D$ are $a=3 \mathrm{~cm} ., b=5 \mathrm{~cm} ., c=7 \mathrm{~cm} .$ and $d=9 \mathrm{~cm}$. Find $\Delta, R, A$ and $D$.

Jay Patel

Numerade Educator

01:53

Problem 238

In a cyclic quadrilateral, prove that the angle between its diagonals is $\sin ^{-1}\left[\frac{2 \sqrt{(s-a)(s-b)(s-c)(s-d)}}{(a c+b d)}\right]$.

Ethan Somes

Numerade Educator

Problem 239

If $A B C D$ be a cyclic quadrilateral, prove that $\tan \frac{B}{2}=\sqrt{\frac{(s-a)(s-b)}{(s-c)(s-d)}}$.

Gio Maya

Numerade Educator

03:30

Problem 240

If $A B C D$ be a cyclic quadrilateral, prove that the product of the segments into which one diagonal is divided by the other diagonal is $\frac{a b c d(a c+b d)}{(a b+c d)(a d+b c)}$.

Jill Tolbert

Numerade Educator

06:05

Problem 241

The sides of a quadrilateral with an inscribed circle are $7,10,5$ and $2 \mathrm{~cm}$. and the sum of a pair of opposite angles is $120^{\circ}$. Find area and radius of inscribed circle.

Andrija Isakov

Numerade Educator

06:13

Problem 242

A quadrilateral $A B C D$ is circumscribed about a circle, prove that $a \sin \frac{A}{2} \sin \frac{B}{2}=c \sin \frac{C}{2} \sin \frac{D}{2}$.

Gaurav Kalra

Numerade Educator

00:20

Problem 243

The sides of a cyclic quadrilateral are $3,3,4$ and $4 \mathrm{~cm} .$. Find the radii of the incircle and circumcircle and the area and the angles.

Ashley High

Numerade Educator

Problem 244

If a cyclic quadrilateral can be circumscribed about another circle, prove that it's area is $\sqrt{a b c d}$ and that the radius of the inscribed circle is $\frac{2 \sqrt{a b c d}}{a+b+c+d}$.

Wen Zheng

Numerade Educator

Problem 245

If a cyclic quadrilateral can be circumscribed about a circle, prove that the angle between its diagonals is $\cos ^{-1}\left(\frac{a c-b d}{a c+b d}\right)$

Gio Maya

Numerade Educator

Problem 246

Let $A B C$ be a triangle with incentre $I$ and inradius $r$. Let $D, E, F$ be the feet of the perpendiculars from $I$ to the sides $B C, C A$ and $A B$ respectively. If $r_{1}, r_{2}$ and $r_{3}$ are the radii of circles inscribed in the quadrilateral $A F I E, B D I F$ and $C E I D$ respectively, prove that $\frac{r_{1}}{r-r_{1}}+\frac{r_{2}}{r-r_{2}}+\frac{r_{3}}{r-r_{3}}=\frac{r_{1} r_{2} r_{3}}{\left(r-r_{1}\right)\left(r-r_{2}\right)\left(r-r_{3}\right)} .$

Wen Zheng

Numerade Educator

00:50

Problem 247

Find the length of the perimeter of a regular decagon which surrounds a circle of radius $12 \mathrm{~cm}$.

Glenn Degamon

Numerade Educator

05:08

Problem 248

Find the length of the side of a regular polygon of 12 sides which is circumscribed to a circle of unit radius.

Suman Saurav Thakur

Numerade Educator

06:00

Problem 249

Find the area of a pentagon, a hexagon, an octagon and a decagon, each being a regular figure of side 1 meter.

Natalie Daly

Numerade Educator

02:08

Problem 250

Find the difference between the areas of a regular octagon and a regular hexagon if the perimeter of each is $24 \mathrm{~cm}$.

Lauren Shelton

Numerade Educator

00:30

Problem 251

A square, whose side is $2 \mathrm{~cm}$., has it's corners cut away so as to form a regular octagon, find it's area.

Amrita Bhasin

Numerade Educator

Problem 252

Compare the areas and perimeters of octagons which are respectively inscribed in and circumscribed to a given circle.

Evan Sun

Numerade Educator

Problem 253

Show that the areas of the hexagon and octagon inscribed to a given circle are as $\sqrt{27}: \sqrt{32}$.

Aman Gupta

Numerade Educator

01:32

Problem 254

If an equilateral triangle and a regular hexagon have the same perimeter, prove that their areas are as $2: 3$.

Aman Gupta

Numerade Educator

03:00

Problem 255

If a regular pentagon and a regular decagon have the same perimeter, prove that their areas are as $2: \sqrt{5}$.

Aman Gupta

Numerade Educator

02:04

Problem 256

Given that the area of a polygon of $n$ sides circumscribed about a circle is to the area of the circumscribed polygon of $2 n$ sides as $3: 2$, find $n$.

Aman Gupta

Numerade Educator

Problem 257

The area of a regular polygon of $n$ sides inscribed in a circle is to that of the same number of sides circumscribing the same circle as $3: 4$. Find the value of $n$.

Aman Gupta

Numerade Educator

02:10

Problem 258

The interior angles of a polygon are in A.P., the least angle is $120^{\circ}$ and the common difference is $5^{\circ} .$ Find the number of sides.

Aman Gupta

Numerade Educator

Problem 259

There are two regular polygons, the number of sides in one being double the number in the other, and an angle of one polygon is to an angle of the other as $9: 8$, find the number of sides of each polygon.

Aman Gupta

Numerade Educator

02:34

Problem 260

Show that there are eleven pairs of regular polygons such that the number of degrees in the angle of one is to the number in the angle of the other as $10: 9$. Find the number of sides in each.

Aman Gupta

Numerade Educator

Problem 261

Prove that the area of the circle and the area of a regular polygon of $n$ sides and of perimeter equal to that of the circle are in the ratio of $\tan \frac{\pi}{n}: \frac{\pi}{n}$.

Aman Gupta

Numerade Educator

01:27

Problem 262

Prove that the sum of the radii of the circles, which are respectively in and circumscribed about a regular polygon of $n$ sides, is $\frac{a}{2} \cot \frac{\pi}{2 n}$, where $a$ is a side of the polygon.

Aman Gupta

Numerade Educator

03:06

Problem 263

Of two regular polygons of $n$ sides, one circumscribed and the other is inscribed in a given circle, prove that the perimeters of the circumscribing polygon, the circle and the inscribed polygon are in the ratio $\sec \frac{\pi}{n}: \frac{\pi}{n} \operatorname{cosec} \frac{\pi}{n}: 1$ and that the areas of the polygons are in the ratio $\cos ^{2} \frac{\pi}{n}: 1$.

Aman Gupta

Numerade Educator