Circle theorem?

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1 Answer
Jan 11, 2018

Please see below.

Explanation:

Let the arc hat(APC) subtend angle /_ABC at thecircumference and angle /_AOC at the center of the circle. Join BO and extend it to meet the circumference at P, as follows.
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As AO=BO=CO each being equal to radius, DeltasABO and BCO are isosceles triangles.

Now in DeltaABO, m/_BAO=m/_ABO and as exterior angle /m_AOP is equal to sum of its interior opposite angles

/_AOP=/_BAO+/_ABO=2/_ABO ................(1)

Similarly in DeltaBOC

/_COP=/_BCO+/_CBO=2/_CBO ................(2)

Adding (1) and (2), we get

/_AOC=/_ABC

Q.E.D.