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Given a triangle $ABC$, and points $D$, $E$, and $F$ on the lines $BC$, $CA$, and $AB$ respectively, then the points $D$, $E$, and $F$ lie in a transversal line if and only if

$$\dfrac{AF}{FB}\dfrac{BD}{DC}\dfrac{CE}{EA}=-1$$

where a length is considered negative if not traversed in the directions $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CA}$, i.e. $AF$ is negative if $F$ lies beyond $A$ and $FB$ is negative if $F$ lies beyond $B$