\[ \begin{align*} \frac{a^4 + b^4 + c^4}{2} &= \frac{1}{2} ( a^4 + b^4 + (-a-b)^4 ) \\[5px] &= \frac{1}{2} ( 2a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + 2b^4 ) \\[5px] &= a^4 + \textcolor{red}{2a^3b} + \textcolor{blue}{3a^2b^2} + \textcolor{green}{2ab^3} + b^4 \\[5px] &= a^4 + \textcolor{red}{a^3b} + \textcolor{blue}{a^2b^2} + \textcolor{red}{a^3b} + \textcolor{blue}{a^2b^2} + \textcolor{green}{ab^3} + \textcolor{blue}{a^2b^2} + \textcolor{green}{ab^3} + b^4 \\[5px] &= a^2(a^2 + ab + b^2) + ab(a^2 + ab + b^2) + b^2(a^2 + ab + b^2) \\[5px] &= (a^2 + ab + b^2)^2 \end{align*} \]

\[ \frac{a^4 + b^4 + c^4}{2} = (a^2 + ab + b^2)^2 = (b^2 + bc + c^2)^2 = (c^2 + ca + a^2)^2 = (ab + bc + ca)^2 \]