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

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

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 a, b, c are in HP then prove that b^2(a-c)^2=2{c^2(b-a)^2+a^2(c-b)^2} .

If the roots of the equation (b-c)x^(2)+(c-a)x+(a-b)=0 are equal,then prove that 2b=a+c

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 the roots of the quadratic equation (a-b) x^(2) + (b - c) x + (c - a) =0 are equal , prove that b +c = 2a

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

If a, b, c, d are in G.P., then prove that: (b-c)^(2)+(c-a)^(2)+(d-b)^(2)=(a-d)^(2)

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