Midsegment converse — a line through the midpoint of one side parallel to another side passes through the midpoint of the third side

Depends on: midsegment theorem (Triangle midsegment theorem), uniqueness of the parallel through an external point (Through a point off a line, exactly one parallel exists (Playfair)).

Statement

Let $\triangle ABC$, and let $D$ be the midpoint of $AB$. Draw $DE\parallel BC$ through $D$ meeting $AC$ at $E$ — then

In other words, in $\triangle ABC$, given the midpoint $D$ of one side together with an auxiliary line parallel to another side, we can immediately pin down the midpoint $E$ of the third side.