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Computable Euclid

Proposition 31c

Theorem

The angle (ACD) in a circular segment (ACD) smaller than a semicircle is an obtuse angle.

Commentary

1. Given a circle centered at O, let AB  be a diameter.
2. Let C be any point on the circumference and connect AC  and BC , constructing ACB.
3. Let D be a point on the circumference, so that arc ACD is shorter than the semicircle. Connect CD  so that ACD is constructed.
4. Then ACD contained in the segment (the area bounded by arc ACD and AD ) is an obtuse angle.
5. Book 3 Propositions 31a covers the case when the segment of the circle is equal to a semicircle. Book 3 Propositions 31b covers the case when the segment of the circle is larger 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.


Computable version


Additional instances


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