Prove the median of right triangle is half the length of the hypotenuse How can I prove that in a right triangle the median which tends to the hypotenuse has length half of that the hypotenuse? I want to show using vector methods or analytic geometry

The converse can be proved to same logic at first.
Let $A$ be half hypotenuse and $B$ be the median vectors
Assume $ |A| = |B|= u $
Taking dot scalar product of sides
$$ ( A + B).(A-B) = |A|^2 - |B|^2= 0 \tag{1} $$
So the sides are perpendicular, must belong to a right triangle.
Next, for the direct proposition by the above relation (1) median length (radius) is half of the hypotenuse.
