If D is the midpoint of the hypotenuse AC of a right triangle ABC. Prove that BD = 1/2 AC.
Since AC is the hypotenuse of right triangle ABC, the the right angle is opposed to the hypotenuse, therefore the angle B measures 90 degrees.
According to enunciation, D is the midpoint of AC, therefore, BD is median in triangle ABC.
We'll draw a parallel to the leg BC, that is passing through the midpoint D.
This parallel line, DE, falls to the midpoint E of the line AB, therefore, the segment DE is the midline of triangle ABC. Since BC is perpendicular to AB, then the parallel line DE is also perpendicular to AB. This means that in triangle ADB, the segment DE is both median and height of triangle ADB. But this thing happens only if the triangle ADB is isosceles.
If triangle ADB is isosceles, then the sides AD = DB (the median and height joins the vertex D with the side AB of triangle ADB).
But AD = DC, since D is the midpoint of hypotenuse AC => BD = AD = DC = AC/2.