Prove that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference

prove that the angle which is subtended by an arc at the centre of a circle is double the size of the angle subtended at any point on the circumference

In ∆ ABC :-

AO = BO = CO

Angle BAO = ANGLE ABO =β & Angle CAO = Angle ACO = α

So Angle AOB  + Angle BAO + Angle ABO = 180

    Angle AOB  = 180 - 2 β              (1)

Similarly  Angle COA  =  180 - 2 α               (2)

Angle AOB  +  Angle COA  + Angle COB = 360

 Angle COB = 360 - 360 + 2 α + 2 β

Angle COB = 2 α + 2 β

Angle BAC = α+  β

Hence Angle COB = 2 Angle BAC