Median to Hypotenuse of Right Triangle equals Half Hypotenuse
Let $AD$ be the median to $BC$.
Then $AD$ is half of $BC$.
Construct $BE$ and $CE$ parallel to $AC$ and $AB$ respectively.
Then by definition $ABEC$ is a parallelogram.
Thus by Quadrilateral is Parallelogram iff Diagonals Bisect each other, $AE$ is also a bisector of $ABEC$.
Thus by definition $ABEC$ is a rectangle.
- $AE = BC$
From above, $D$ is the midpoint of both $AE$ and $BC$.
Thus $AD = BD$ and hence the result.