Trapezoid Midsegment Theorem

Assume on contrary that $MN\not\parallel AB$.

Draw parallel lines from M and N.

By Toolkit 109 — Triangle Proportionality Theorem and its Converse in $\triangle DAB$,

Therefore, E is the midpoint of DB.

By Toolkit 109 — Triangle Proportionality Theorem and its Converse in $\triangle DBC$,

Therefore, E' is the midpoint of DB.

By Toolkit 109 — Triangle Proportionality Theorem and its Converse in $\triangle DAB$,

Similarly,