Let the trapezoid be $ABCD$ with bases $BC$ and $AD$ with $AD>BC$.
1.Constructing midpoint of base $AD$
We can construct point $P$ intersection of lines $AB$ and $CD$. We also can construct point $Q$ the intersection of $AC$ and $BD$. It's a well-known fact that $PQ$ passes through the midpoint of $AD$. Let $M$ be the intersection.
2. Constructing the midline
Since $BCDM$ is a parallelogram, $BM\parallel CD$. The midline of triangle $APD$ passes through $M$ and is parallel to $PD$, so $BM$ Is the midline of triangle $APD$. Let $AC\cap BM=K$. We can see that $K$ is the midpoint of $AC$. Analogously, let $L=CM\cap BD$ we have that $L$ is the midpoint of $BD$. The midline of trapezoid passes through the midpoints of diagonals, from this follows, that $KL$ is the midline of the trapezoid.
