If I , I 1 , I 2 & I 3 be respectively the centers of the incircle and the three escribed circles of a triangle A B C , then prove that area of ∆ I 1 I 2 I 3 = 8 R 2 cos A 2 cos B 2 cos C 2 = a b c 2 r , where r is inradius, R is circumradius and a , b & c are the length of the sides opposite to angle A , B & C respectively.

Let s be the semiperimeter, S be the area of triangle and r 1 , r 2 & r 3 be exradii. We know, I 2 I 3 = a cosec A 2 & A I 1 = r 1 cosec A 2 Now, area of ∆ I 1 I 2 I 3 = 1 2 I 2 I 3 × A I 1 = 1 2 a cosec A 2 × r 1 cosec A 2 Using a = 2 R sin A & r 1 = 4 R sin A 2 cos B 2 cos C 2 , we get = 1 2 2 R sin A cosec A 2 × 4 R sin A 2 cos B 2 cos C 2 cosec A 2 = 8 R 2 cos A 2 cos B 2 cos C 2 … i Apply the formula, cos A 2 = s s - a b c , cos B 2 = s s - b a c & cos C 2 = s s - c a b = 8 R 2 s s - a b c s s - b a c s s - c a b = 8 R 2 × s a b c × S = 8 × a b c 4 S 2 × s a b c × S = 8 × a b c 16 S × s = a b c 2 S × s = a b c 2 S s Using r = S s = a b c 2 r Hence, area of ∆ I 1 I 2 I 3 = 8 R 2 cos A 2 cos B 2 cos C 2 = a b c 2 r .

MEDIUM

JEE Main

IMPORTANT

Earn 100

If $I, I_{1}, I_{2} & I_{3}$ be respectively the centres of the incircle and the three escribed circles of a triangle $A B C$ , prove that: $I_{2} I_{3}^{2} = 4 R (r_{2} + r_{3})$ .

Important Questions on Properties of Triangle

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 8

I, I_{1}, I_{2} & I_{3}

be respectively the centres of the incircle and the three escribed circles of a triangle

A B C

, prove that:

∠ I_{3} I_{1} I_{2} = \frac{B + C}{2}

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 9

I, I_{1}, I_{2} & I_{3}

be respectively the centres of the incircle and the three escribed circles of a triangle

A B C

, prove that:

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}

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 10

I, I_{1}, I_{2} & I_{3}

be respectively the centers of the incircle and the three escribed circles of a triangle

A B C

, then prove that area of

∆ 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}

, where

r

is inradius,

R

is circumradius and

a, b & c

are the length of the sides opposite to angle

A, B & C

respectively.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 11

I, I_{1}, I_{2} & I_{3}

be respectively the centres of the incircle and the three escribed circles of a triangle

A B C

, prove that:

\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}

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 12

In triangle

A B C

, prove that

I O^{2} = R^{2} (3 - 2 \cos A - 2 \cos B - 2 \cos C)

where

O & I

are circum-centre and in-centre of triangle

A B C

respectively and

R

is circum-radius.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 13

In a triangle $A B C$ , prove that $I P^{2} = 2 r^{2} - 4 R^{2} \cos A \cos B \cos C$ , where $P & I$ are orthocentre and incentre of triangle $A B C$ and $r & R$ are inradius and circumradius of given triangle.

HARD

JEE Main

IMPORTANT

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

In a triangle $A B C$ , prove that $O G^{2} = R^{2} - \frac{1}{9} (a^{2} + b^{2} + c^{2})$ , where $a, b, c$ are length of sides of triangle opposite to angle $A, B & C$ , respectively and $O & G$ are circum-centre and centroid of triangle and $R$ is circum radius of given triangle.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 15

In triangle

A B C

, prove that area of

∆ I O P = 2 R^{2} \sin \frac{B - C}{2} \sin \frac{C - A}{2} \sin \frac{A - B}{2}

, where

I, O & P

are in-centre, circum-centre and orthocentre of triangle

A B C

, and

R

is circum-radius of triangle.

If I, I1, I2 & I3 be respectively the centres of the incircle and the three escribed circles of a triangle ABC, prove that: I2I32=4R(r2+r3).

Important Questions on Properties of Triangle

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If $I, I_{1}, I_{2} & I_{3}$ be respectively the centres of the incircle and the three escribed circles of a triangle $A B C$ , prove that: $I_{2} I_{3}^{2} = 4 R (r_{2} + r_{3})$ .