A. Seen from DE//AC and DF//AB, the quadrilateral AEDF is a parallelogram.

B From DE//AC, DF//AB and angle BAC=90 degrees, it can be concluded that the quadrilateral AEDF is a rectangle.

D. Connect EF, if AD is perpendicular to BC and AB=AC, DE//AC, DF//AB,

It can be proved that EF//BC, then AD is also perpendicular to EF, then the quadrilateral AEDF is a diamond.

C. If there is only the bisecting angle BAC of AD, it cannot be guaranteed that AD is perpendicular to EF, so the quadrilateral AEDF is not necessarily a diamond.