It is a constant term of Laurent polynomial $f/x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1}$, this guy changes its sign after we replaces $x_i$ to $1/x_i$, thus the result.

This coefficient $L$ is a constant term of the Laurent polynomial $g(x_1,x_2,x_3)=f(x_1,x_2,x_3)/x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1}$, this guy $g$ satisfies $g(x_1,x_2,x_3)=-g(1/x_1,1/x_2,1/x_3)$, thus $L=-L$.