If I be the centre of the incircle of a triangle A B C , prove that A I = r cosec A 2 . where r is the radius of the incircle.

Let perpendicular from I on the side A C is E . So, E I = r Now, in ∆ A I E , sin A 2 = E I A I ⇒ A I E I = cosec A 2 Or A I = E I cosec A 2 = r cosec A 2 ∵ E I = r

If I is incentre of triangle A B C , prove that: I A · I B · I C = a b c tan A 2 tan B 2 tan C 2 , where a , b , c are length of sides opposite to angle A , B & C .

We know that I A = r cosec A 2 I B = r cosec B 2 and I C = r cosec C 2 ∴ I A = 4 R sin A 2 sin B 2 sin C 2 cos A 2 cos A 2 cosec A 2 = a sin B 2 sin C 2 cos A 2 cosec A 2 Similarly, I B = b sin C 2 sin A 2 cos B 2 cosec B 2 and I C = c sin A 2 sin B 2 cos C 2 cosec C 2 ∴ I A · I B · I C = a b c · sin A 2 sin B 2 sin C 2 cos A 2 cos B 2 cos C 2 = a b c · tan A 2 · tan B 2 · tan C 2

E M B I B E

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVI>Q 19

MEDIUM

JEE Main

IMPORTANT

Earn 100

Prove that in triangle $A B C$ , $(r_{1} - r) (r_{2} - r) (r_{3} - r) = 4 R r^{2}$ , where $R$ is radius of circumcircle and $r$ is radius of incircle, $r_{1}, r_{2} & r_{3}$ are the radii of the excircles.

Important Questions on Properties of Triangle

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVI>Q 20

Prove that in triangle

A B C

,

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

, where

R

is circumradius of the circumcircle and

r

is the inradius of incircle,

a, b & c

are the sides of the triangle

△ A B C

.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVI>Q 21

Prove that in triangle

A B C

,

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

, where

R

is radius of circumcircle and

r_{1}, r_{2} & r_{3}

are the radius of excircles and

r

is radius of incircle,

a, b & c

are the sides of the triangle.

HARD

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVI>Q 22

Prove that in triangle

A B C

,

r^{2} + r_{1}^{2} + r_{2}^{2} + r_{3}^{2} = 16 R^{2} - a^{2} - b^{2} - c^{2}

, where

R

is radius of circumcircle,

r_{1}, r_{2} & r_{3}

are the radius of excircles,

r

is radius of incircle and

a, b & c

are the sides of the triangle.

MEDIUM

JEE Main

IMPORTANT

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

If

I

be the centre of the incircle of a triangle

A B C

, prove that

A I = r cosec \frac{A}{2}

. where

r

is the radius of the incircle.

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 2

If $I$ is incentre of triangle $A B C$ , prove that: $I A \cdot I B \cdot I C = a b c \tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}$ , where $a, b, c$ are length of sides opposite to angle $A, B & C$ .