Given:
\[{{m}_{1}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{1}}~=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\]
\[{{m}_{2}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{2}}~=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\]
and,
\[{{m}_{3}}x\text{ }\text{ }y\text{ }+\text{ }{{c}_{3}}~=\text{ }0\text{ }\ldots \text{ }\left( 3 \right)\]
since, the three lines are concurrent.
Now,
\[{{m}_{1}}\left( -{{c}_{3}}~+\text{ }{{c}_{2}} \right)\text{ }+\text{ }1\left( {{m}_{2}}{{c}_{3}}-{{m}_{3}}{{c}_{2}} \right)\text{ }+\text{ }{{c}_{1}}\left( -{{m}_{2}}~+\text{ }{{m}_{3}} \right)\text{ }=\text{ }0\]
\[{{m}_{1}}\left( {{c}_{2}}-{{c}_{3}} \right)\text{ }+\text{ }{{m}_{2}}\left( {{c}_{3}}-{{c}_{1}} \right)\text{ }+\text{ }{{m}_{3}}\left( {{c}_{1}}-{{c}_{2}} \right)\text{ }=\text{ }0\]
∴ The required condition is:
\[{{m}_{1}}\left( {{c}_{2}}-{{c}_{3}} \right)\text{ }+\text{ }{{m}_{2}}\left( {{c}_{3}}-{{c}_{1}} \right)\text{ }+\text{ }{{m}_{3}}\left( {{c}_{1}}-{{c}_{2}} \right)\text{ }=\text{ }0\]