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

EASY

Earn 100

What is pedal triangle?

Important Questions on Theorems of Concurrency

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

. If

∠ B E D = k °

then find

k

.

EASY

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

In

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

is

MEDIUM

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

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

.

MEDIUM

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

What is pedal triangle?

∆ A B C

is an obtuse triangle of which

∠ B

is obtuse, construct its pedal triangle.

HARD

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

Circumradius of

Δ A B C

is

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 __________

EASY

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

If

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>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 $\frac{R}{2} .$

EASY

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

With usual notations, consider that

I_{1}, I_{2} & I_{3}

are ex-centres of

∆ A B C

. If the ratio of circum-radius of triangle

I_{1} I_{2} I_{3}

to the circum-radius of pedal triangle of triangle

A B C

is

K : 1

, then the value of

K

is

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.

HARD

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

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

is

HARD

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

The area of an acute triangle

A B C

is

∆ .

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

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>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

∆ 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>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 =

EASY

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

With usual notations consider that

I_{1}, I_{2}

and

I_{3}

are excentres of triangle

A B C

. If the ratio of circumradius of triangle

I_{1} I_{2} I_{3}

to the circumradius of pedal triangle of triangle

A B C

is

K : 1

, then the value of

K

is

HARD

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

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

.

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

, prove that

\frac{E F}{a \cos A} = \frac{F D}{b \cos B} = \frac{D E}{c \cos C}

.

HARD

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

Find the ratio of area of

△ D E F

to area of

△ A B C

.