Question

Prove that 1/(a-b)² +1(b-c)²+1/(c-a)² is a perfect square.​

Answer 1

Answer:

Answer

Given,

a

2

+b

2

+c

2

−ab−bc−ca

multiply and divide by 2

=

2

2

×(a

2

+b

2

+c

2

−ab−bc−ca)

=

2

a

2

−2ab+b

2

+b

2

−2bc+c

2

+c

2

−2ac+a

2

=

2

(a−b)

2

+(b−c)

2

+(c−a)

2

square of a number is always greater than or equal to zero.

∴(a−b)

2

+(b−c)

2

+(c−a)

2

≥0

and

(a−b)

2

+(b−c)

2

+(c−a)

2

=0 when a=b=c

Hence, a

2

+b

2

+c

2

−ab−bc−ca is always non negative