This is of great importance for its consequences, of which the two following are the principal:—
Prop. 21. The angles in the same segment of a circle are equal to one another;
Prop. 22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
Further consequences are:—
Prop. 23. On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another;
Prop. 24. Similar segments of circles on equal straight lines are equal to one another.
The problem Prop. 25. A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop. 1, III., in § 27.
§ 34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs. They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.
The theorems are:—
Prop. 26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences;