Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 3

MEDIUM

JEE Main

IMPORTANT

Earn 100

If $I, I_{1}, I_{2}$ and $I_{3}$ be respectively the centres of the incircle and the three escribed circles of a triangle $A B C$ . Prove that: $A I_{1} = r_{1} cosec \frac{A}{2}$ .

Important Questions on Properties of Triangle

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 4

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 I_{1} = a \cdot \sec \frac{A}{2}

, where

a, b, c

are sides of triangle

A B C

opposite to angle

A, B, C

.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 5

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} = a \cdot cosec \frac{A}{2}

.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 6

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 I_{1} \cdot I I_{2} \cdot I I_{3} = 16 R^{2} r

.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 7

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})

.

MEDIUM

JEE Main

IMPORTANT

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

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_{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

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 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

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 \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

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:

\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}

.