Proposition 31b
Theorem
The angle (∠BCE ) in a circular segment (BCE ) greater than a semicircle is an acute angle.
Commentary
1. Given a circle centered at O, let AB be a diameter.
2. Let C be any point on the circumference and connectAC and BC , constructing ∠ACB .
3. Let E be a point on the circumference, so that arc ECB is longer than the semicircle. ConnectCE so that ∠BCE is constructed.
4. Then∠BCE contained in the segment (the area bounded by arc BCE and BE ) is an acute angle.
5. Book 3 Propositions 31a covers the case when the segment of the circle is equal to a semicircle. Book 3 Propositions 31c covers the case when the segment of the circle is smaller than a semicircle.
2. Let C be any point on the circumference and connect
3. Let E be a point on the circumference, so that arc ECB is longer than the semicircle. Connect
4. Then
5. Book 3 Propositions 31a covers the case when the segment of the circle is equal to a semicircle. Book 3 Propositions 31c covers the case when the segment of the circle is smaller than a semicircle.
Original statement
ἐν κύκλῳ ἡ μὲν ἐν τῷ ἡμικυκλίῳ γωνία ὀρθή ἐστιν, ἡ δὲ ἐν τῷ μϵίζονι τμήματι ἐλάττων ὀρθῆς, ἡ δὲ ἐν τῷ ἐλάττονι τμήματι μϵίζων ὀρθῆς: καὶ ἔτι ἡ μὲν τοῦ μϵίζονος τμήματος γωνία μϵίζων ἐστὶν ὀρθῆς, ἡ δὲ τοῦ ἐλάττονος τμήματος γωνία ἐλάττων ὀρθῆς.
English translation
In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; and further the angle of the greater segment is greater than a right angle, and the angle of the less segment less than a right angle.
