One property that is needed here is the one which says that if you add one row (or column) of a determinant to another, then the value of the determinant is unchanged.
So for example if you add each of the top two rows to the bottom row, then you get
$\displaystyle \begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} = \begin{vmatrix}a&b&c\\b&c&a\\a+b+c&a+b+c&a+b+c\end {vmatrix}$ .
The next property that comes in handy is the one saying that if you multiply each element in a row (or column) by a constant, then the value of the determinant is multiplied by that constant. Or, to put it another way, if each element in a row (or column) has the same factor, then you can take that factor outside the determinant. Thus
$\displaystyle \begin{vmatrix}a&b&c\\b&c&a\\a+b+c&a+b+c&a+b+c\end {vmatrix} = (a+b+c)\begin{vmatrix}a&b&c\\b&c&a\\1&1&1\end{vmat rix}$ .
That gets the factor a+b+c for you. Now use similar procedures to deal with the rest of the problem.