∠ADO=∠DAO=x, ∠ODB=∠DBO=y (isosceles ∆s AOD and BOD).
∠AOD=180-2x, ∠BOD=180-2y, ∠AOB=360-(180-2x+180-2y),
∠AOB=2(x+y). But x+y=∠ADB.
∠ACO=∠CAO=w, ∠OCB=∠OBC=z (isosceles ∆s COA and COB).
∠AOC=180-2w, ∠BOC=180-2z,
∠AOB=∠BOC-∠AOC=180-2z-(180-2w)=2(w-z).
But w-z=∠ACB.
Therefore ∠AOB=2(∠ADB)=2(∠ACB) and ∠ADB=∠ACB, that is,
Angles at the circumference of a circle subtended by the same chord or arc are congruent.