ABC is equilateral triangle .

Prove AO= 2/3  AD

First of all remember that for an equilateral triangle,  the medians, angle bisectors and altitudes coincide for an equilateral triangle because of it's symmetry.
Consider BO joined. In triangle OBD, 
<OBD = 30 degree.
sin 30 = ¹/2   = OD / OB .But, OD = OA 
 So, ¹/2   = OD / OA 
Therefore,OA = 2 OD
ie: AO = 2/3 AD