Angles subtended by an arc

An arc is a connected piece of a circle. Two points A and B cut the circle into two arcs: the bigger is the major arc, the smaller the minor arc.

The angle an arc subtends at the centre O is the angle you sweep going from OA to OB along that arc. Less than 180° means a minor arc; more than 180°, a major arc.

Theorem 9: the angle at the centre is double the angle at the circle. The angle an arc subtends at the centre is twice the angle it subtends at any point D on the circle outside the arc.

How the proof works. Join D through the centre C. This makes two isosceles triangles (two radii each). In each, the exterior angle at C equals the two equal base angles added, so it is twice the base angle. Add the two parts and you get the theorem. The book checks both cases, depending on where DC meets the circle.

So the angle at D does not depend on where D is, as long as D is on the circle and outside the arc: ∠AEB = ∠ADB = ∠AFB = ½ ∠ACB. Points inside or outside the circle see AB at different angles; only points on the circle agree.

Corollary: the angle in a semicircle is 90°. If AB is a diameter, the arc not containing D subtends 180° at the centre, so ∠ADB = 90°.