b+c-a a a a b c
b c+a-b b = b c a
c c a+b-c c a b
The brute force approach:
Calculate $\displaystyle \left| \begin{array}{ccc}
b + c - a & a & a \\
b & c + a - b & b \\
c & c & a + b - c \end{array} \right|$.
Calculate $\displaystyle \left| \begin{array}{ccc}
a & b & c \\
b & c & a \\
c & a & b \end{array} \right|$.
Alternatively, you could do row operations on $\displaystyle \left| \begin{array}{ccc}
a & b & c \\
b & c & a \\
c & a & b \end{array} \right|$.