Math, asked by ronakrathore12345, 6 months ago

o is any point in the of a ∆ ABC .prove that ( oa+ob+oc)>half (ab+bc+ca)

Answers

Answered by DilipChaurasia
0

Answer:

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Answered by prabhjotkaur83
0

Step-by-step explanation:

Let O be a inner point of a triangle ABC. We are to prove that

O

A

+

O

B

+

O

C

<

A

B

+

B

C

+

C

A

*Construction"

B

O

is produced to intersect AB at D.

For

Δ

A

B

D

A

B

+

A

D

>

B

D

A

B

+

A

D

>

B

O

+

O

D

...

.

.

[

1

]

For

Δ

C

O

D

C

D

+

O

D

>

O

C

...

...

[

2

]

Adding [1] and [2] we get

A

B

+

A

D

+

C

D

+

O

D

>

B

O

+

O

D

+

O

C

A

B

+

A

C

+

O

D

>

B

O

+

O

D

+

O

C

A

B

+

A

C

>

O

B

+

O

C

...

.

.

(

3

)

Similarly

A

B

+

B

C

>

O

A

+

O

C

...

.

.

(

4

)

And

A

C

+

B

C

>

O

A

+

O

B

...

.

.

(

5

)

Adding (3),(4)and (5) we get

2

(

A

C

+

B

C

+

C

A

)

>

2

(

O

A

+

O

B

+

O

C

)

O

A

+

O

B

+

O

C

<

A

B

+

B

C

+

C

A

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