$\begin{array}{l} \dfrac{{\left( {a + b - c} \right)}}{{ab}} - \dfrac{{\left( {b + c - a} \right)}}{{bc}} - \dfrac{{\left( {a + c - b} \right)}}{{ac}} = 0\\ \Leftrightarrow \dfrac{{c\left( {a + b - c} \right) - a\left( {b + c - a} \right) - b\left( {a + c - b} \right)}}{{abc}} = 0\\ \Leftrightarrow \dfrac{{ac + bc - {c^2} - ab - ac + {a^2} - ab - bc + {b^2}}}{{abc}} = 0\\ \Leftrightarrow \dfrac{{{a^2} - 2ab + {b^2} - {c^2}}}{{abc}} = 0\\ \Leftrightarrow \dfrac{{{{\left( {a - b} \right)}^2} - {c^2}}}{{abc}} = 0\\ \Leftrightarrow \dfrac{{\left( {a - b - c} \right)\left( {a - b + c} \right)}}{{abc}} = 0\\ \Rightarrow \left[ \begin{array}{l} a = b + c\\ a - b + c = 0\left( 2 \right) \end{array} \right.\\ \left( 2 \right) \to a - b + c = 0 \Rightarrow \dfrac{{a + c - b}}{{ac}} = \dfrac{{b - c + c - b}}{{abc}} = 0 \end{array}$
