If a square + b square + c square - ab -bc -ca = 0 Then Prove that a=b=c
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a2+b2+c2-ab-bc-ac=0
multiply and divide by 2
2a2+2b2+2c2-2ab-2bc-2ac
reorganizing
a2+b2-2ab+b2+c2-2bc+c2+a2-2ac+a2+b2+c2=0
we get
(a-b)²+(b-c)²+(a-c)²+a²+b²+c²=0
as a2+b2+c2≠0
hence a=b=c as a-b=0 a=c
b-c=0 b=c
a-c=0 a=c
multiply and divide by 2
2a2+2b2+2c2-2ab-2bc-2ac
reorganizing
a2+b2-2ab+b2+c2-2bc+c2+a2-2ac+a2+b2+c2=0
we get
(a-b)²+(b-c)²+(a-c)²+a²+b²+c²=0
as a2+b2+c2≠0
hence a=b=c as a-b=0 a=c
b-c=0 b=c
a-c=0 a=c
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