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.).