prove that `a^2+b^2+c^2-ab-bc-ca=1/2[(a-b)^2+(b-c)^2+(c-a)^2]`

Prove that: 2a^(2)+2b^(2)+2c^(2)-2ab-2bc-2ca=[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]

Prove that: 2a^(2)+2b^(2)+2c^(2)-2ab-2bc-2ca=(a-b)^(2)+(b-c)^(2)+(c-a)^(2)

Prove that: 2x^(2)+2b^(2)+2c^(2)-2ab-2bc-2ca=[(a-b)^(2)+(b-c)^(2)+(c-a)^(2)]

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Prove that |[a^2, a^2-(b-c)^2, bc],[b^2, b^2-(c-a)^2, ca],[c^2, c^2-(a-b)^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

Prove that |[(b+c)^2, a^2, bc],[(c+a)^2, b^2, ca],[(a+b)^2, c^2, ab]|=(a-b)(b-c)(c-a)(a+b+c)(a^2+b^2+c^2)

If a+b+c=2 and ab+bc+ca=1 then find (a+b)^2+(b+c)^2+(c+a)^2

Prove that |{:(a^(2), a^(2)-(b-c)^(2), bc), (b^(2), b^(2)-(c-a)^(2), ca), (c^(2), c^(2)-(a-b)^(2), ab):}|= (a^(2)+b^(2)+c^(2))(a-b)(b-c)(c-a)(a+b+c)

If a,b,c are in G.P.then prove that (a^(2)+ab+b^(2))/(bc+ca+ab)=(b+a)/(c+b)

Prove that |[(a+b)^(2),ca,bc],[ca,(b+c)^(2),ab],[bc,ab,(c+a)^(2)]|=2abc(a+b+c)^(3)