Question

in triangle abc find the value of r1/bc + r2/ca +r3/ab​

Answer 1

Answer:

+

ca

r

2

+

ab

r

3

=

bc

stan

2

A

+

ac

stan

2

B

+

ab

stan

2

C

=

abc

s

[atan

2

A

+btan

2

B

+ctan

2

C

]

Now using a=2RsinA=2R.2sin

2

A

.cos

2

A

=4Rsin

2

A

.cos

2

A

, and similarly using the formula for b and c, we get

bc

r

1

+

ca

r

2

+

ab

r

3

=

abc

s

.4R[sin

2

2

A

+sin

2

2

B

+sin

2

2

C

]

Using r=

s

△

and R=

4△

abc

, we get

abc

s

.4R=

r

1

.

.

⇒

bc

r

1

+

ca

r

2

+

ab

r

3

=

r

1

[

2

1−cosA

+

2

1−cosB

+

2

1−cosC

]

.

=

r

1

[

2

3

−

2

cosA+cosB+cosC

]

Now using cosA+cosB+cosC=1+

R

r

, we get

bc

r

1

+

ca

r

2

+

ab

r

3

=

r

1

[1−

2R

r

]

.

⇒

bc

r

1

+

ca

r

2

+

ab

r

3

=

r

1

−

2R

1