Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 27

EASY

JEE Main

IMPORTANT

Earn 100

Prove that the product of the distance of the in-centre from the angular points of a triangle is $4 R r^{2}$ .

Important Questions on Properties of Triangle

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 28

The triangle

∆ D E F

, circumscribes the three escribed 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}

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 29

If a circle be drawn touching the inscribed and circumscribed circle of a triangle and the side

B C

, externally, prove that its radius is

\frac{∆}{a} \tan^{2} \frac{A}{2}

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 31

If

∆_{0}

be the area of the triangle formed by joining the points of contact of the inscribed circle with the sides of the given triangle, whose area is

∆

, and

∆_{1}, ∆_{2}

and

∆_{3}

the corresponding areas for the escribed circles, prove that

∆_{1} + ∆_{2} + ∆_{3} - ∆_{0} = 2 ∆

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 32

If the bisector of the angle of a triangle $A B C$ meet the opposite sides in $A^{'}, B^{'}$ and $C^{'}$ , prove that the ratio of the areas of the triangles $A^{'} B^{'} C^{'} & A B C$ is $2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} : \cos \frac{A - B}{2} \cos \frac{B - C}{2} \cos \frac{C - A}{2}$ .

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 33

Through the angular points of a triangle are drawn straight line which make the same angle

α

with the area of opposite sides of the triangle; Prove that the area of the triangle formed by them is to the original triangle as

4 \cos^{2} α : 1

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 34

Two circles, of radii

a

and

b

, cut each other at an angle

θ

. Prove that the length of the common chord is

\frac{2 a b \sin θ}{\sqrt{a^{2} + b^{2} + 2 a b \cos θ}}

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 35

Three equal circles touch one another; find the radius of the circles which touches all three.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 36

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