Midline and Median of Triangle Bisect Each Other
Let $\triangle ABC$ be a triangle.
Then $AF$ and $DE$ bisect each other.
Construct the midlines $DF$ and $EF$.
Then by the Midline Theorem $DF \parallel AE$ and $EF \parallel AD$.
Thus by Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel, $\Box ADFE$ is a parallelogram.
By construction, $AF$ and $DE$ are the diagonals of $\Box ADFE$.
The result follows from Diameters of Parallelogram Bisect each other.