help me prove this

From the question,BD is the diameter and AC is a tangent.
Therefore,angle BDA=angle BDC=90°
Also, AD is median.Thus by SAS congruence,triangle ABD is congruent to triangle ACD.
Therefore AB=BC
Thus triangle ABC is equilateral.
If we join ED, angle DEB is 90°[angle in semicircle]
angle A=60°
cos A=AE/AD=1/2
→AE=AD/2
→AE=(AC/2)/2
→AE=AC/4
Thus proved
There are many other ways to do this sum as well....