" 13.Prove that "a^(2)+b^(2)+c^(2)-ab-bc-ca" is always non-negative for all values of "a,b" and "c

Prove that a^2+b^2+c^2-a b-b c-c a is always non-negative for all values of a ,b a nd c.

Prove that a^2+b^2+c^2-a b-b c-c a is always non-negative for all values of a ,\ b\ a n d\ c

Prove that a^(2)+b^(2)+c^(2)=2(ab cos C +bc cosA +ca cos B)

If a^(2)+b^(2)+c^(2)-ab-bc-ca=0, prove that a=b=c

If :a^(2)+b^(2)+c^(2)-ab-bc-ca=0, prove that a=b=c

Prove that |[b^2+c^2,ab,ac],[ab,c^2+a^2,bc],[ca,cb,a^2+b^2]| is always positive for real a, b and c.

The value of determinant |(bc-a^(2),ac-b^(2),ab-c^(2)),(ac-b^(2),ab-c^(2),bc-a^(2)),(ab-c^(2),bc-a^(2),ac-b^(2))| is a)always non -negative b)always non-positive c)always zero d)can't say anything

If a!=b!=c then prove that (a^(2)+b^(2)+c^(2))/(ab+bc+ca)>1

If a^2 + b^2 + c^2 -ab-bc-ca = 0 , then prove that a=b=c

Given that a,b,c are distinct real numbers such that expressions ax^(2)+bx+c,bx^(2)+cx+aandcx^(2)+ax+b are always non-negative.Prove that the quantity (a^(2)+b^(2)+c^(2))/(ab+bc+ca) can never lie inn (-oo,1)uu[4,oo)