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

MEDIUM

Earn 100

If in $Δ A B C$ , the line joining the circum-centre and the in-centre is parallel to $B C$ , then-

50% studentsanswered this correctly

Important Questions on Properties of Triangle

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Let

D, E, F

be points on the sides

B C, C A, A B

of a triangle

A B C,

respectively. Suppose

A D, B E, C F

are concurrent at

P .

\frac{P F}{P C} = \frac{2}{3}, \frac{P E}{P B} = \frac{2}{7}

and

\frac{P D}{P A} = \frac{m}{n}

where

m, n

are positive integers with

gcd (m, n) = 1 .

Find

(m + n)

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

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.

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Value of

\frac{a}{p} + \frac{b}{q} + \frac{c}{r}

is equal to (where

a, b, c

are length of sides

BC

CA

and

AB

respectively)

EASY

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

For a triangle

A B C, R = \frac{5}{2}

and r=1. Let D,E and F be the feet of the perpendicular from incentre I to BC, CA and AB, respectively. Then the value of

\frac{(I A) (I B) (I C)}{(I D) (I E) (I F)}

is equal to _________

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

∆ A B C

, it is given that the distance between the circumcentre

(O)

and orthocentre

(H)

R \sqrt{1 - 8 \cos A \cos B \cos C} .

Q

is the mid-point of

O H

, then

A Q

EASY

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

A B C

is an acute - angled triangle with circumcentre

' O'

and orthocentre

H .

A O = A H

, then angle

A

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

If the length of median

A A_{1}, B B_{1}

and

C C_{1}

Δ A B C

are

m_{a}, m_{b}

and

m_{c},

respectively. Then,

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

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.

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

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 $A, B & C$ .

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Consider a

∆ A B C

and

D, E

and

F

are the foot of the perpendicular drawn from the vertices

A, B

and

C

respectively. Let

H

be the orthocentre of the triangle

A B C

. Then the value of

\frac{\cos A \cdot \cos B \cdot \cos C}{\cos^{2} A + \cos^{2} B + \cos^{2} C}

is:

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

H

is the orthocentre of

ΔABC, R =

circumradius and

P = AH + BH + CH,

then

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Δ A B C, A C = 3, B C = 4

and medians

A D

and

B E

are perpendicular, then find the value of

A B^{2} .

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

If the distances of the vertices of a triangle from the point of contact of the incircle with the sides be

α, β, γ

then

r

is equal to (where

r =

inradius)

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

In a ∆ABC, the line joining circumcenter to the incenter is parallel to BC, then value of cos B + cos C is

EASY

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Prove that the product of the distance of the in-centre from the angular points of a triangle is

4 R r^{2}

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Consider a

∆ A B C

and

D, E

and

F

are the foot of the perpendicular drawn from the vertices

A, B

and

C

respectively. Let

H

be the orthocentre of the triangle

A B C

. Then the value of

R

is:

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Δ A B C, b = 12

units,

c = 5

units and

Δ = 30 sq .

units. If

d

is the distance between vertex

A

and incentre of the triangle, then the value of

d^{2}

EASY

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

Cotyledons are also called-

HARD

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

In triangle

A B C

prove that area of

∆ I P G = \frac{4}{3} R^{2} \sin \frac{B - C}{2} \sin \frac{C - A}{2} \sin \frac{A - B}{2}

, where

I, P & G

are incentre, circumcentre and orthocentre of triangle and

R

is circumradius of given triangle.

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Distances of the Centres of a Triangle from Its Vertices and Sides

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}

If in ΔABC, the line joining the circum-centre and the in-centre is parallel to BC, then-

Important Questions on Properties of Triangle

Learn and Prepare for any exam you want !

If in $Δ A B C$ , the line joining the circum-centre and the in-centre is parallel to $B C$ , then-