Answer:

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If a

2

+b

2

+c

2

−ab−bc−ca=0, prove that a=b=c.

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Solution

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Consider, a

2

+b

2

+c

2

–ab–bc–ca=0

Multiply both sides with 2, we get

2(a

2

+b

2

+c

2

–ab–bc–ca)=0

⇒ 2a

2

+2b

2

+2c

2

–2ab–2bc–2ca=0

⇒ (a

2

–2ab+b

2

)+(b

2

–2bc+c

2

)+(c

2

–2ca+a

2

)=0

⇒ (a–b)

2

+(b–c)

2

+(c–a)

2

=0

Since the sum of square is zero then each term should be zero

⇒ (a–b)

2

=0,(b–c)

2

=0,(c–a)

2

=0

⇒ (a–b)=0,(b–c)=0,(c–a)=0

⇒ a=b,b=c,c=a

∴ a=b=c.

Answer:

If a²+b²+c² = ab+bc+ca, then (c+a)/b = 2. Thus, a = b = c.