And when the straight lines containing an angle cut off some circumference, the angle is said to stand upon that (circumference).
We say that the inscribed angle $\alpha$ stands upon the arc $AB$, if it is contained by the straight lines joining the ends of the arc $AB$ with any point on the circumference of a segment of the circle.
Proofs: 1 2 3 4 5
Propositions: 6 7 8