but there is a geometrical proof i got.
Drop perpendicular from B to AD and call the foot of the perpendicular as K. Let AC meet BD at L.
then |AK|=|KD| and |AL|=|LC|.
Let AL and BK meet at G. G is the centroid since AL and BK are medians.
To prove: |AD|/|AC| < 2/3
that is, |AK|/|AL| < 2/3.
Now |AK|<|AG| since AG is the hypotenuse of triangle-AGK.(AGK is a right angled triangle).
so |AK|/|AL| < |AG|/|AL|.
we know that |AG|/|AL|=2/3 from elementary geometry because G is the centroid. which proves the result.