I was going through the proof of a theorem which says, "The angle subtended by a chord at the center of a circle is twice the angle subtended by it elsewhere on the circle."

This was proved by considering three cases: when the arc forming the chord was:

1) a minor arc

2) a major arc

3) a semi circle

I realized that the following scenario has been avoided, which is what I'm trying to prove:

Prove that: Angle AOB = 2 * Angle APB

You see, the textbook very cleverly takes the point P above the point O, which allows joining O and P, and the problem is easily solved by using the property of exterior angles.

Is this interpretation right? If not, why? If yes, how is it to be proved?

Thanks in advance!