It's correct.

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.