The inscribed angle

An angle whose vertex lies on a circle and whose sides cut the circle is called an inscribed angle; the arc between its sides is the arc it subtends. Theorem: an inscribed angle is measured by half of the arc it subtends, that is ∠ABC = ½ ∪AC. The proof considers whether the centre lies on a side, inside the angle or outside it; in the first case the exterior angle of the isosceles triangle AOB is used. Corollary 1: all inscribed angles subtending the same arc are equal. Corollary 2: an inscribed angle subtending a diameter (a semicircle) is a right angle. Also, an inscribed angle is half the central angle on the same arc.