Euclid · Elements §3.prop.31

Angles in Semicircles and Segments Compared to a Right Angle

Passage 51 of 316 · Greek

Summary

This proposition proves the properties of angles in a circle, showing that the angle in a semicircle is a right angle, that in a segment greater than a semicircle is less than a right angle, and that in a segment less than a semicircle is greater than a right angle, along with the properties of the angles of the segments themselves.

§3.prop.31ἐν 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.

In a circle the angle in the semicircle is a right angle, that in the greater segment less than a right angle, and that in the 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.

ἔστω κύκλος ὁ ΑΒΓΔ, διάμετρος δὲ αὐτοῦ ἔστω ἡ ΒΓ, κέντρον δὲ τὸ Ε, καὶ ἐπεζεύχθωσαν αἱ ΒΑ, ΑΓ, ΑΔ, ΔΓ· λέγω, ὅτι ἡ μὲν ἐν τῷ ΒΑΓ ἡμικυκλίῳ γωνία ἡ ὑπὸ ΒΑΓ ὀρθή ἐστιν, ἡ δὲ ἐν τῷ ΑΒΓ μείζονι τοῦ ἡμικυκλίου τμήματι γωνία ἡ ὑπὸ ΑΒΓ ἐλάττων ἐστὶν ὀρθῆς, ἡ δὲ ἐν τῷ ΑΔΓ ἐλάττονι τοῦ ἡμικυκλίου τμήματι γωνία ἡ ὑπὸ ΑΔΓ μείζων ἐστὶν ὀρθῆς.

Let ΑΒΓΔ be a circle, let ΒΓ be its diameter, and Ε the center, and let ΒΑ, ΑΓ, ΑΔ, ΔΓ be joined; I say that the angle ΒΑΓ in the semicircle ΒΑΓ is a right angle, the angle ΑΒΓ in the segment ΑΒΓ greater than the semicircle is less than a right angle, and the angle ΑΔΓ in the segment ΑΔΓ less than the semicircle is greater than a right angle.

ἐπεζεύχθω ἡ ΑΕ, καὶ διήχθω ἡ ΒΑ ἐπὶ τὸ Ζ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΒΕ τῇ ΕΑ, ἴση ἐστὶ καὶ γωνία ἡ ὑπὸ ΑΒΕ τῇ ὑπὸ ΒΑΕ. πάλιν, ἐπεὶ ἴση ἐστὶν ἡ ΓΕ τῇ ΕΑ, ἴση ἐστὶ καὶ ἡ ὑπὸ ΑΓΕ τῇ ὑπὸ ΓΑΕ· ὅλη ἄρα ἡ ὑπὸ ΒΑΓ δυσὶ ταῖς ὑπὸ ΑΒΓ, ΑΓΒ ἴση ἐστίν.

Let ΑΕ be joined, and let ΒΑ be carried through to Ζ. And since ΒΕ is equal to ΕΑ, the angle ΑΒΕ is also equal to the angle ΒΑΕ. Again, since ΓΕ is equal to ΕΑ, the angle ΑΓΕ is also equal to the angle ΓΑΕ; therefore the whole angle ΒΑΓ is equal to the two angles ΑΒΓ, ΑΓΒ.

ἐστὶ δὲ καὶ ἡ ὑπὸ ΖΑΓ ἐκτὸς τοῦ ΑΒΓ τριγώνου δυσὶ ταῖς ὑπὸ ΑΒΓ, ΑΓΒ γωνίαις ἴση· ἴση ἄρα καὶ ἡ ὑπὸ ΒΑΓ γωνία τῇ ὑπὸ ΖΑΓ· ὀρθὴ ἄρα ἑκατέρα· ἡ ἄρα ἐν τῷ ΒΑΓ ἡμικυκλίῳ γωνία ἡ ὑπὸ ΒΑΓ ὀρθή ἐστιν.

But the angle ΖΑΓ, which is exterior to the triangle ΑΒΓ, is also equal to the two angles ΑΒΓ, ΑΓΒ; therefore the angle ΒΑΓ is also equal to the angle ΖΑΓ; therefore each is a right angle; therefore the angle ΒΑΓ in the semicircle ΒΑΓ is a right angle.

καὶ ἐπεὶ τοῦ ΑΒΓ τριγώνου δύο γωνίαι αἱ ὑπὸ ΑΒΓ, ΒΑΓ δύο ὀρθῶν ἐλάττονές εἰσιν, ὀρθὴ δὲ ἡ ὑπὸ ΒΑΓ, ἐλάττων ἄρα ὀρθῆς ἐστιν ἡ ὑπὸ ΑΒΓ γωνία· καί ἐστιν ἐν τῷ ΑΒΓ μείζονι τοῦ ἡμικυκλίου τμήματι.

And since in the triangle ΑΒΓ the two angles ΑΒΓ, ΒΑΓ are less than two right angles, and the angle ΒΑΓ is a right angle, the angle ΑΒΓ is less than a right angle; and it is in the segment ΑΒΓ greater than the semicircle.

καὶ ἐπεὶ ἐν κύκλῳ τετράπλευρόν ἐστι τὸ ΑΒΓΔ, τῶν δὲ ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν, καί ἐστιν ἡ ὑπὸ ΑΒΓ ἐλάττων ὀρθῆς· λοιπὴ ἄρα ἡ ὑπὸ ΑΔΓ γωνία μείζων ὀρθῆς ἐστιν· καί ἐστιν ἐν τῷ ΑΔΓ ἐλάττονι τοῦ ἡμικυκλίου τμήματι.

And since ΑΒΓΔ is a quadrangle in a circle, and the opposite angles of quadrangles in circles are equal to two right angles, and the angle ΑΒΓ is less than a right angle, the remaining angle ΑΔΓ is greater than a right angle; and it is in the segment ΑΔΓ less than the semicircle.

λέγω, ὅτι καὶ ἡ μὲν τοῦ μείζονος τμήματος γωνία ἡ περιεχομένη ὑπό τῆς ΑΒΓ περιφερείας καὶ τῆς ΑΓ εὐθείας μείζων ἐστὶν ὀρθῆς, ἡ δὲ τοῦ ἐλάττονος τμήματος γωνία ἡ περιεχομένη ὑπό τῆς ΑΔ περιφερείας καὶ τῆς ΑΓ εὐθείας ἐλάττων ἐστὶν ὀρθῆς.

I say that also the angle of the greater segment, namely that contained by the circumference ΑΒΓ and the straight line ΑΓ, is greater than a right angle, and the angle of the less segment, namely that contained by the circumference ΑΔ and the straight line ΑΓ, is less than a right angle.

καί ἐστιν αὐτόθεν φανερόν.

And it is manifest at once.

ἐπεὶ γὰρ ἡ ὑπὸ τῶν ΒΑ, ΑΓ εὐθειῶν ὀρθή ἐστιν, ἡ ἄρα ὑπὸ τῆς ΑΒΓ περιφερείας καὶ τῆς ΑΓ εὐθείας περιεχομένη μείζων ἐστὶν ὀρθῆς.

For, since the angle contained by the straight lines ΒΑ, ΑΓ is a right angle, the angle contained by the circumference ΑΒΓ and the straight line ΑΓ is greater than a right angle.

πάλιν, ἐπεὶ ἡ ὑπὸ τῶν ΑΓ, ΑΖ εὐθειῶν ὀρθή ἐστιν, ἡ ἄρα ὑπὸ τῆς ΓΑ εὐθείας καὶ τῆς ΑΔ περιφερείας περιεχομένη ἐλάττων ἐστὶν ὀρθῆς.

Again, since the angle contained by the straight lines ΑΓ, ΑΖ is a right angle, the angle contained by the straight line ΓΑ and the circumference ΑΔ is less than a right angle.

ἐν κύκλῳ ἄρα ἡ μὲν ἐν τῷ ἡμικυκλίῳ γωνία ὀρθή ἐστιν, ἡ δὲ ἐν τῷ μείζονι τμήματι ἐλάττων ὀρθῆς, ἡ δὲ ἐν τῷ ἐλάττονι μείζων ὀρθῆς, καὶ ἔτι ἡ μὲν τοῦ μείζονος τμήματος μείζων ὀρθῆς, ἡ δὲ τοῦ ἐλάττονος τμήματος ἐλάττων ὀρθῆς· ὅπερ ἔδει δεῖξαι.

Therefore in a circle the angle in the semicircle is a right angle, that in the greater segment less than a right angle, and that in the 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; which was to be proved.

Notes

¦5¦ἡ μὲν τοῦ μείζονος τμήματος γωνία — Note the distinction between 'the angle in the segment' (ἡ ἐν τῷ τμήματι γωνία, i.e., a rectilineal angle/inscribed angle standing on the chord with its vertex on the circumference) and 'the angle of the segment' (ἡ τοῦ τμήματος γωνία, i.e., a mixed angle contained by the arc and the chord). The former is a standard rectilineal angle, while the latter involves a curve.
¦20¦ὅλη ἄρα ἡ ὑπὸ ΒΑΓ δυσὶ ταῖς ὑπὸ ΑΒΓ, ΑΓΒ ἴση ἐστίν — Since ΒΕ = ΕΑ and ΓΕ = ΕΑ (radii of the circle), the base angles of isosceles triangles are equal, yielding angle ΑΒΕ = angle ΒΑΕ and angle ΑΓΕ = angle ΓΑΕ. Since the point Ε lies on the diameter ΒΓ, angle ΑΒΕ is identical to angle ΑΒΓ, and angle ΑΓΕ to angle ΑΓΒ. Thus, the whole angle ∠ΒΑΓ (= ∠ΒΑΕ + ∠ΓΑΕ) is equal to ∠ΑΒΓ + ∠ΑΓΒ.
¦40¦ἡ ὑπὸ τῶν ΒΑ, ΑΓ εὐθειῶν — When the preposition ὑπό is followed by a genitive plural ('the straight lines ΒΑ, ΑΓ', τῶν εὐθειῶν), it denotes a rectilineal angle 'contained' by the two lines (angle ΒΑΓ). In contrast, when singular genitives are connected by καί ('the circumference ΑΒΓ and the straight line ΑΓ'), it represents a mixed angle 'contained' by the arc and the line. This grammatical variation clearly distinguishes rectilineal angles from mixed angles.

Cite this passage

Euclid, Elements §3.prop.31. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:3.prop.31

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