Question

In triangle ABC, a = 5, 6 = 6, c = 7. I is the incenter of triangle ABC. P is the circumcenter of triangle IBC.R is the circumcenter of triangle IAB.Q is the circumcenter of triangle ICA. Find circumradius of triangle PQR.

Answer 1

Answer:

In △IBC

∠IBC=

2

B

and ∠ICB=

2

C

⇒∠BIC=π−(

2

B+C

)

From sine rule:

sin∠BIC

BC

=2P

1

⇒2P

1

=

sin(

2

B+C

)

a

=asec

2

A

Similarly in ΔICA and ΔIAB, we get 2P

2

=bsec

2

B

and 2P

3

=csec

2

C

respectively.

Now, P

1

P

2

P

3

=

8cos

2

A

cos

2

B

cos

2

C

abc

=

cos

2

A

cos

2

B

cos

2

C

R

3

sinAsinBsinC

⇒P

1

P

2

P

3

=2R

2

(4Rsin

2

A

sin

2

B

sin

2

C

)=2R

2

r

Ans: A,B,C,D