Question

if a^2+b^2+c^2-ab-bc-ca=0prove that a=b,b=c,c=a or a=b=c.

Answer 1

a^2+b^2+c^2-ab-bc-ca=0
a^2+(a)^2+(a)^2-ab-bc-ca=0 [ a=b=c]
3a^2-ab-bc-ca=0
3a^2-a(a)-(a)(a)-(a)a=0 [a=b=c]
3a^2-a^2-a^2-a^2=0
3a^2-3a^2=0
0=0
hence proved