Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 18

HARD

JEE Main

IMPORTANT

Earn 100

$D E F$ 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 $A B C$ .

Important Questions on Properties of Triangle

MEDIUM

JEE Main

IMPORTANT

Plane Trigonometry Part 1>Chapter 14 - Properties of Triangle>Examples XXXVII>Q 18

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

JEE Main

IMPORTANT

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

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

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

A B C;

prove that its sides are

4 R \cos \frac{A}{2}, 4 R \cos \frac{B}{2} & 4 R \cos \frac{C}{2}

, where

R

is the circum-centre of triangle

A B C

.

MEDIUM

JEE Main

IMPORTANT

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

O_{1} O_{2} O_{3}

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

A B C

; prove that its angles are

\frac{π}{2} - \frac{A}{2}, \frac{π}{2} - \frac{B}{2}

and

\frac{π}{2} - \frac{C}{2}

.

HARD

JEE Main

IMPORTANT

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

∆ 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

JEE Main

IMPORTANT

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

∆ D E F

is the triangle formed by joining the point of contact of the in-circle with the sides of the triangle

∆ A B C

, prove that its sides are

2 r \cos \frac{A}{2}, 2 r \cos \frac{B}{2}

and

2 r \cos \frac{C}{2}

.

MEDIUM

JEE Main

IMPORTANT

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

∆ D E F

is the triangle formed by joining the points of contact of the in-circle with the sides of the triangle

∆ A B C

, prove that its angle are

\frac{π}{2} - \frac{A}{2}, \frac{π}{2} - \frac{B}{2}

and

\frac{π}{2} - \frac{C}{2}

.

MEDIUM

JEE Main

IMPORTANT

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

∆ D E F

is the triangle formed by joining the point of contact of the in-circle with the sides of the triangle

∆ A B C

, prove that its area is

\frac{2 ∆^{3}}{a b c}

, i.e.,

\frac{1}{2} \frac{r}{R} ∆

.

MEDIUM

JEE Main

IMPORTANT

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

D, E

and

F

are the middle points of the sides of the triangle

∆ A B C

, prove that centroid of the triangle

∆ D E F

is the same as that of

∆ A B C

and that its orthocenter is the circumcenter of

∆ A B C

.