Let G be the mid-point of EC. Draw DG.
Rt. tris. DCE similar to ADE. Since G and F are mid-points of CE and DE, respectively. So Rt. tris. AFE similar to DGE, so angles FAE = GDE. subtract them from DAE=CDE, we have angles DAF=CDG.
Because DG // BE, so angles CDG=DBE, so angles DBE=DAF.
Let AF and BE intersect at M, we can conclude A, O, D and B are cyclic.
Finally angles AOB=ADB=90 degree. Q.E.D.