[tex]\det A = \left|\begin{array}{ccc}a&b&c\\b&c&a\\c&a&b\end{array}\right|= \left|\begin{array}{ccc}a&a+b+c&c\\b&a+b+c&a\\c&a+b+c&b\end{array}\right| = \\ \\ = (a+b+c)\cdot \left|\begin{array}{ccc}a&1&c\\b&1&a\\c&1&b\end{array}\right| = (a+b+c)\cdot \left|\begin{array}{ccc}a-b&0&c-a\\b&1&a\\c-b&0&b-a\end{array}\right|= \\ \\ = (a+b+c)\cdot \Big[(a-b)(b-a)-(c-b)(c-a)\Big][/tex]
