From figure, Let C is the mid-point of chord AB.
To prove: D is the mid-point of arc AB.
Now, In ΔOAC and ΔOBC
OA = OB [Radius of circle]
OC = OC [Common]
AC = BC [C is the mid-point of chord AB (given)]
So, by SSS condition:
ΔOAC ≅ ΔOBC
So, ∠AOC = ∠BOC (BY CPCT)