Draw the diagonal $BD$, then draw the lines $l_{1}$ and $l_{2}$ which are parallel to $BD$ and pass through $A$ and $C$ respectively. WLOG the length between $BD$ and $l_{1}$ is greater than or equal to the length between $BD$ and $l_{2}$. Draw $AB$ to cut $l_{2}$ at $E$. So we get $AB\geq BE$ and $S_{ADE}=S_{ABCD}$. Let $F$ be the midpoint of $AE$. We know that $AB\geq BE$, so $F$ is on segment $AB$ and $S_{AFD}=\frac{1}{2}S_{ABCD}$. Thus, the line $DF$ is the line the problem requires.