The First Six Books of the Elements of Euclid

by Euclid; John Casey

PROP. XX.—Theorem.
The angle (AOB) at the centre (O) of a circle is double the angle (ACB) at the circumference standing on the same arc.

Dem.—Join CO, and produce it to E. Then because OA is equal to OC, the angle ACO is equal to OAC; but the angle AOE is equal to the sum of the two angles OAC, ACO. Hence the angle AOE is double the angle ACO. In like manner the angle EOB is double the angle OCB. Hence (by adding in figs. (α), (β), and subtracting in (γ)), the angle AOB is double of the angle ACB.

Cor.—If AOB be a straight line, ACB will be a right angle—that is, the angle in a semicircle is a right angle (compare xxxi.).

PROP. XXI.—Theorem.
The angles (ACB, ADB) in the same segment of a circle are equal.

Dem.—Let O be the centre. Join OA, OB. Then the angle AOB is double of the angle ACB [xx.], and also double of the angle ADB. Therefore the angle ACB is equal to the angle ADB.

The following is the proof of the second part—that is, when the arc AB is not greater than a semicircle, without using angles greater than two right angles:—

Let O be the centre. Join CO, and produce it to meet the circle again in E. Join DE. Now since O is the centre, the segment ACE is greater than a semicircle; hence, by the first case, fig. (α), the angle ACE is equal to ADE. In like manner the angle ECB is equal to EDB. Hence the whole angle ACB is equal to the whole angle ADB.