You visited us 1 times! Enjoying our articles? Unlock Full Access!

The Mid-Point Theorem

'O' is a poin...

Question

$^{'} O^{'}$ is a point inside of $△ A B C$ . The bisector of $∠ A O B, ∠ B O C, ∠ C O A$ meet the sides $A B, B C$ and $C A$ in points $D, E$ and $F$ respectively, then $A D . B E . C F =$

D B . E C . F A

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

A D . E C . F A

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D B . B E . F A

No worries! We‘ve got your back. Try BYJU‘S free classes today!

D B . B E . C F

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A $D B . E C . F A$
In $△ A O B$ $O D$ is the bisector of $∠ A O B$
$\frac{O A}{O B} = A D / D B - - - - - - (1)$
In $△ B O C$ $O E$ is the bisector of $∠ B O C$
$\frac{O B}{O C} = \frac{B E}{E C} - - - - - - (2)$
In $△ C O A$ $O F$ is the bisector of $∠ C O A$
$\frac{O E}{O A} = \frac{C F}{F A} - - - - - - - (3)$
$\frac{O A}{O B} \times \frac{O B}{O C} \times \frac{O C}{O A} = \frac{A D}{D B} \times \frac{B E}{E C} \times \frac{C F}{F A}$
$⊥ = \frac{A D}{D B} \times \frac{B E}{E C} \times \frac{C F}{F A}$
$D B \times E C \times F A = A D \times B E \times C F$

Suggest Corrections

Similar questions

O is any point inside a triangle ABC. The bisector of $∠$ AOB, $∠$ BOC and $∠$ COA meet the sides AB. BC and CA in point D. E and F respectively. Show that AD $\times$ BE $\times$ CF = DB $\times$ EC $\times$ FA. [4 MARKS]

Q. In given figure O is any point inside a triangle ABC. The bisector of

∠ A O B

∠ B O C

and

∠ C O A

meet the sides AB, BC and CA in point D,E and F respectively. Show that

A D \times B E \times C F = D B \times E C \times F A

Y

is any point inside a triangle

A B C

. The bisector of

∠ A Y B

∠ B Y C

and

∠ C Y A

meet the sides,

A B, B C

and

C A

in point

D, E

and

F

respectively.
Then,

A D \times B E

D B \times F A

Q. If

D, E, F

are the midpoints of the sides

B C, C A, A B,

respectively, of a triangle

A B C

and

O

is any point, then

Join BYJU'S Learning Program

′O′ is a point inside of △ABC. The bisector of ∠AOB,∠BOC,∠COA meet the sides AB,BC and CA in points D,E and F respectively, then AD.BE.CF=

$^{'} O^{'}$ is a point inside of $△ A B C$ . The bisector of $∠ A O B, ∠ B O C, ∠ C O A$ meet the sides $A B, B C$ and $C A$ in points $D, E$ and $F$ respectively, then $A D . B E . C F =$