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

EASY

Earn 100

What is pedal triangle?

Important Questions on Theorems of Concurrency

HARD

Mathematics>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

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>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

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}

HARD

Mathematics>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

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>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

D E F

is the pedal triangle of the equilateral triangle

A B C

. Find

∠ B E D

HARD

Mathematics>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

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:

MEDIUM

Mathematics>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

If, in

Δ A B C, r_{3} = r_{1} + r_{2} + r,

then

∠ A + ∠ B =

HARD

Mathematics>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

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>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

∆ 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

HARD

Mathematics>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

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>Geometry>Theorems of Concurrency>Concurrent Lines and Related Theorems

The triangle

∆ D E F

, circumscribes the three escribed circles of the triangle

∆ A B C