If : `a^2+b^2+c^2-a b-b c-c a=0` , prove that `a=b=c`

If a/b =b/c and a,b, c gt 0 , then prove that (a+b)^2/(b+c)^2 = (a^2 +b^2)/(b^2 +c^2)

If b/(a+b) = (a+c-b)/(b+c-a) = (a+b+c)/(2a+b+2c)("where " a + b + c ne 0) , then prove that a/2 = b/3 = c/4 .

If a + b + c = 0 , then prove that (2a^2)/(a^2-b^2-c^2)+(2b^2)/(b^2-c^2-a^2)+(2c^2)/(c^2-a^2-b^2)=3

Prove that |b c-a^2c a-b^2a b-c^2-b c+c a+a bb c-c a+a bb c+c a-a b(a+b)(a+c)(b+c)(b+a)(c+a)(c+b)|=3.(b-c)(c-a)(a-b)(a+b+c)(a b+b c+c a)

If a + b + c = 0 , prove that a^(4) + b^(4) + c^(4) = 2(b^(2)c^(2)+c^(2)a^(2)+a^(2)b^(2)) = 1//2 (a^(2) + b^(2) + c^(2))^(2)

If (b^2+c^2-a^2)/(2b c),(c^2+a^2-b^2)/(2c a),(a^2+b^2-c^2)/(2a b) are in A.P. and a+b+c=0 then prove that a(b+c-a),b(c+a-b),c(a+b-c) are in A.P.

If a!=b!=c\ a n d\ |{:(a, b, c), (a^2,b^2,c^2), (b+c, c+a, a+b):}|=0 then a+b+c=0 b. a b+b c+c a=0 c. a^2+b^2+c^2=a b+b c+c a d. a b c=0

If a!=b!=c\ and\ |{:(a, b, c), (a^2,b^2,c^2), (b+c, c+a, a+b):}|=0 then A. a+b+c=0 B. a b+b c+c a=0 C. a^2+b^2+c^2=a b+b c+c a D. a b c=0

If a!=b!=c\ a n d\ |{:(a, b, c), (a^2,b^2,c^2), (b+c, c+a, a+b):}|=0 then a+b+c=0 b. a b+b c+c a=0 c. a^2+b^2+c^2=a b+b c+c a d. a b c=0

Prove that: |[b, c-a^2,c] ,[a-b^2,a b-c^2,c ],[a-b^2,a ,b-c^2b c-a^2a b-c^2b c-a^2c a-b^2]|=|[a, b, c],[ b ,c ,a],[ c, a ,b]|^2 .