$$$
\begin{array}{l}
\left| \begin{array}{ccc} a&b&c \\ c&a&b \\ b&c&a \end{array} \right|
= \left| \begin{array}{ccc} a+b+c&b&c \\ c+a+b&a&b \\ b+c+a&c&a \end{array} \right|
\\\\ = (a+b+c) \left| \begin{array}{ccc} 1&b&c \\ 1&a&b \\ 1&c&a \end{array} \right|
\\\\ = (a+b+c) \left| \begin{array}{ccc} 1&b&c \\ 0&a-b&b-c \\ 0&c-b&a-c \end{array} \right|
\\\\ = (a+b+c) \left| \begin{array}{cc} a-b&b-c \\ c-b&a-c \end{array} \right|
\\\\ = (a+b+c)\big((a-b)(a-c) - (b-c)(c-b)\big)
\\\\ = (a+b+c)(a^2 - ab - ac + bc + b^2 - 2bc + c^2)
\\\\ = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca)
\end{array}
$$$