Cho a^2 + b^2 +c^2 = ab + bc + ca. Chứng minh a^3+b^3+c^3=3abc

$\begin{array}{l}a^3 + b^3 + c^3 – 3abc\\ = (a^3 + b^3) + c^3 – 3abc\\ = (a + b)^3 – 3ab(a + b) + c^3 – 3abc\\ = [(a + b)^3 + c^3] – 3ab(a + b + c)\\ = (a + b + c)[(a + b)^2 – (a + b)c + c^2] – 3ab(a + b + c)\\ = (a + b + c)(a^2 + 2ab + b^2 – ac – bc + c^2) – 3ab(a + b + c)\\ = (a+ b + c)(a^2 + 2ab + b^2 – ac – bc + c^2 – 3ab)\\ = (a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)\\ = 0 \quad (Do \,a^2 +b^2 + c^2 = ab + bc + ca)\\ \Rightarrow a^3 + b^3 + c^3 = 3abc\end{array}$