Question
Question: In a triangle ABC, r<sup>2</sup> + r<sub>1</sub><sup>2</sup> + r<sub>2</sub><sup>2</sup> + r<sub>3</...
In a triangle ABC, r2 + r12 + r22 + r32 + a2 + b2 + c2 is equal to–
A
4R2
B
8R2
C
12R2
D
16R2
Answer
16R2
Explanation
Solution
Here, r2 + r12 + r22 + a2 + b2 + c2
= r2 + (r1 + r2 + r3)2 – 2(r1r2 + r2r3 + r1r3)
+ (a + b + c)2 – 2 (ab + bc + ca)
= r2 + (4R + r)2 – 2s2 + 4s2 – 2 (ab + bc + ca)
= 2r2 + 16R2 + 8rR + 2s2 – 2 (ab + bc + ca)
= 16R2 + 
+ 
+ 2s2 – 2 (ab + bc + ca)
= 16R2 + 
+ 2abc + 2s2 –2(ab + bc + ca)
= 16R2 + 2s2 – s. 4s + 2 (ab + bc + ca) + 2s2 –2(ab + bc + ca)
= 16R2.