Proposition 31a
Theorem
Alternate name(s): Thales' theorem.
In a circle, the angle (∠BCA ) in a semicircle is a right angle.
Commentary
1. Given a circle centered at O, let AB be a diameter.
2. Let C be any point on the circumference, connectAC and BC , constructing ∠ACB .
3. Then∠ACB is contained in the semicircle segment (or is contained by the arc) and is a right angle.
4. This proposition is also known as Thales' theorem.
5. Book 3 Propositions 31b covers the case when the segment of the circle is larger than 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, connect
3. Then
4. This proposition is also known as Thales' theorem.
5. Book 3 Propositions 31b covers the case when the segment of the circle is larger than 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.
