TLS: Angle at center is twice angle at circumference

We have learnt that the above approach can be used to show the angle property of angle at center is twice angle at circumference.

Thanks to Junyang and Sharon's question, how will we facilitate the discussion for the explanation/proof of this property if it is of the case below? 



Hint: think about angle in semicircle. Some of you may think that this is circular argument as angle in semicircle is derived from this angle property. However angle in semicircle can also be explained by two isosceles triangles.