Question

if a 2 +b 2 +c 2 -ab-bc-ca = 0 then prove that a=b=c.

Answer 1

Given that:
a^2 + b^2 + c^2 -ab-bc-ac=0
Multiplication with 2 gives;
2a^2 + 2b^2 + 2c^2 -2ab-2bc-2ac=0
On rearrangement of terms, this gives–
(a-b)^2+(b-c)^2+(c-a)^2=0
As we know that square of any number is greater than or equal to zero(0);
Above equation gives–
a=b, b=c, c=a;
=> a=b=c
Proved!