D E F is the pedal triangle of A B C ; prove that the radius of its circum-circle is R 2 , where R is circum-radius of triangle A B C .

Radius of circum-circle of ∆ D E F = a cos A 2 sin 180 ° - 2 A Apply the condition: sin π - θ = sin θ = a cos A 2 sin 2 A = a cos A 4 sin A cos A = a 4 sin A = R 2

HARD

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Statement - 1 : If R be the circumradius of a $Δ ABC$ then circumradius of its excentral $Δ I_{1} I_{2} I_{3}$ is 2R. ${{where I}_{1}, I_{2,} I_{3} are excentre ∆ ABC}$

Statement - 2 : If circumradius of a triangle be R, then circumradius of its pedal triangle is $\frac{R}{2}$ .

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Important Questions on Properties of Triangle

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

D, E . F

are the foot of thee perpendiculars from vertices

A, B, C

to sides

B C, C A, A B

respectively, and

H

is the orthocentre of acute angled triangle

A B C

, where

a, b, c

are the sides of triangle

A B C

, then

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

Circumradius of

Δ A B C

3 c m

and its area is

6 c m^{2}

. If

D E F

is the triangle formed by feet of the perpendicular drawn from

A, B

and

C

on the sides

B C, C A

and

A B

, respectively, then the perimeter of

Δ D E F

(in cm) is __________

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

A_{1} B_{1} C_{1}

is the triangle formed by joining the feet of the perpendiculars drawn from

A B C

upon the opposite sides; in like manner

A_{2} B_{2} C_{2}

is the triangle obtained by joining the feet of the perpendiculars from

A_{1}, B_{1}

and

C_{1}

on the opposite sides and so on. Find the values on the angles

A_{n}, B_{n}

and

C_{n}

in the

n^{t h}

of these triangles.

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

D E F

is the pedal triangle of the equilateral triangle

A B C

. If

∠ B E D = k °

then find

k

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

Find the ratio of area of

△ D E F

to area of

△ A B C

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

If, in

Δ A B C, r_{3} = r_{1} + r_{2} + r,

then

∠ A + ∠ B =

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

A D, B E

and

C F

are the perpendiculars from the angular points of a

∆ A B C

upon the opposite sides. The perimeters of the

∆ D E F

and

∆ A B C

are in the ratio-

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

D E F

is the pedal triangle of

A B C;

prove that the radius of its incircle is

2 R \cos A \cos B \cos C

, where

R

is circumradius of triangle

A B C

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

The area of an acute triangle

A B C

∆ .

If the area of its pedal triangle is

p

, where

\cos B = \frac{2 p}{Δ}

and

\sin B = \frac{2 \sqrt{3} p}{Δ} .

The value of

8 (\cos^{2} A \cos B + \cos^{2} C)

is:

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

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}

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

D E F

is the pedal triangle of the equilateral triangle

A B C

. Find

∠ B E D

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

If in a

Δ A B C, A D, B E

and

C F

are the altitudes and

R

is the circumradius, then the radius of the circumcircle of triangle

D E F

EASY

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

O

is the ortho-center of a triangle

∆ A B C

, prove that the radii of the circles circumscribing the triangle

∆ B O C, ∆ C O A, ∆ A O B

and

∆ A B C

are all equal.

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

STATEMENT- $1$ : If $R$ be the circumradius of a $∆ A B C,$ then the circumradius of its excentral $Δ I_{1} I_{2} I_{3}$ is $2 R .$

STATEMENT- $2$ : If the circumradius of a triangle be $R,$ then the circumradius of its pedal triangle is $\frac{R}{2} .$

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

D E F

is the pedal triangle of

A B C;

prove that the radius of its circum-circle is

\frac{R}{2}

, where

R

is circum-radius of triangle

A B C

EASY

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

Δ A B C,

A D, B E

and

C F

are the altitudes and

R

is the circumradius. Then the radius of the circle through

∆ D E F

EASY

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

Cotyledons are also called-

HARD

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

∆ I_{1} I_{2} I_{3}

is the triangle formed by the centres of the described circles of the triangle

∆ A B C

, prove that its area is

2 R s

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

A K, B L

and

C M

are the perpendiculars from the angular points of a triangle

∆ A B C

upon the opposite sides; prove that the diameters of the circumcircles of the triangles

∆ A M L, ∆ B K M

and

∆ C K L

, are respectively

a \cot A, b \cot B

and

c \cot C

, and that the perimeters of the triangles

∆ K L M

and

∆ A B C

, are in the ratio

r : R

MEDIUM

Mathematics>Trigonometry>Properties of Triangle>Excentral Triangle and Pedal Triangle

What is pedal triangle?

∆ A B C

is an obtuse triangle of which

∠ B

is obtuse, construct its pedal triangle.

Statement - 1 : If R be the circumradius of a Δ ABC then circumradius of its excentral Δ I 1 I 2 I 3 is 2R. { where I1,I2, I3 are excentre ∆ ABC} Statement - 2 : If circumradius of a triangle be R, then circumradius of its pedal triangle is R 2 .

Important Questions on Properties of Triangle

Learn and Prepare for any exam you want !

Statement - 1 : If R be the circumradius of a $Δ ABC$ then circumradius of its excentral $Δ I_{1} I_{2} I_{3}$ is 2R. ${{where I}_{1}, I_{2,} I_{3} are excentre ∆ ABC}$

Statement - 2 : If circumradius of a triangle be R, then circumradius of its pedal triangle is $\frac{R}{2}$ .