Humanitext Reader

Euclid · Elements §3.prop.20-3.prop.22

Inscribed Angle Theorem and Cyclic Quadrilaterals

Passage 47 of 316 · Greek

Summary

Proposition 20 proves that the central angle is double the circumferential angle subtending the same arc, and Proposition 21 shows that angles in the same segment of a circle are equal. Proposition 22 demonstrates that the opposite angles of a cyclic quadrilateral are equal to two right angles.

§3.prop.20ἐν κύκλῳ ἡ πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ τῆς πρὸς τῇ περιφερείᾳ, ὅταν τὴν αὐτὴν περιφέρειαν βάσιν ἔχωσιν αἱ γωνίαι.
In a circle the angle at the center is double of the angle at the circumference, when the angles have the same circumference as base.
ἔστω κύκλος ὁ ΑΒΓ, καὶ πρὸς μὲν τῷ κέντρῳ αὐτοῦ γωνία ἔστω ἡ ὑπὸ ΒΕΓ, πρὸς δὲ τῇ περιφερείᾳ ἡ ὑπὸ ΒΑΓ, ἐχέτωσαν δὲ τὴν αὐτὴν περιφέρειαν βάσιν τὴν ΒΓ· λέγω, ὅτι διπλασίων ἐστὶν ἡ ὑπὸ ΒΕΓ γωνία τῆς ὑπὸ ΒΑΓ. ἐπιζευχθεῖσα γὰρ ἡ ΑΕ διήχθω ἐπὶ τὸ Ζ. ἐπεὶ οὖν ἴση ἐστὶν ἡ ΕΑ τῇ ΕΒ, ἴση καὶ γωνία ἡ ὑπὸ ΕΑΒ τῇ ὑπὸ ΕΒΑ·
Let ΑΒΓ be a circle, and let the angle at its center be ΒΕΓ, and the angle at the circumference ΒΑΓ, and let them have the same circumference ΒΓ as base; I say that the angle ΒΕΓ is double of the angle ΒΑΓ.
αἱ ἄρα ὑπὸ ΕΑΒ, ΕΒΑ γωνίαι τῆς ὑπὸ ΕΑΒ διπλασίους εἰσίν.
For let ΑΕ be joined and drawn through to Ζ.
ἴση δὲ ἡ ὑπὸ ΒΕΖ ταῖς ὑπὸ ΕΑΒ, ΕΒΑ·
Since then ΕΑ is equal to ΕΒ, the angle ΕΑΒ is also equal to the angle ΕΒΑ; therefore the angles ΕΑΒ, ΕΒΑ are double of the angle ΕΑΒ.
καὶ ἡ ὑπὸ ΒΕΖ ἄρα τῆς ὑπὸ ΕΑΒ ἐστι διπλῆ.
And the angle ΒΕΖ is equal to the angles ΕΑΒ, ΕΒΑ; therefore the angle ΒΕΖ is also double of the angle ΕΑΒ.
διὰ τὰ αὐτὰ δὴ καὶ ἡ ὑπὸ ΖΕΓ τῆς ὑπὸ ΕΑΓ ἐστι διπλῆ.
For the same reason the angle ΖΕΓ is also double of the angle ΕΑΓ.
ὅλη ἄρα ἡ ὑπὸ ΒΕΓ ὅλης τῆς ὑπὸ ΒΑΓ ἐστι διπλῆ.
Therefore the whole angle ΒΕΓ is double of the whole angle ΒΑΓ.
Κεκλάσθω δὴ πάλιν, καὶ ἔστω ἑτέρα γωνία ἡ ὑπὸ ΒΔΓ, καὶ ἐπιζευχθεῖσα ἡ ΔΕ ἐκβεβλήσθω ἐπὶ τὸ Η. ὁμοίως δὴ δείξομεν, ὅτι διπλῆ ἐστιν ἡ ὑπὸ ΗΕΓ γωνία τῆς ὑπὸ ΕΔΓ, ὧν ἡ ὑπὸ ΗΕΒ διπλῆ ἐστι τῆς ὑπὸ ΕΔΒ·
Let it be bent back again, and let there be another angle ΒΔΓ, and let ΔΕ be joined and produced to Η.
λοιπὴ ἄρα ἡ ὑπὸ ΒΕΓ διπλῆ ἐστι τῆς ὑπὸ ΒΔΓ. ἐν κύκλῳ ἄρα ἡ πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ τῆς πρὸς τῇ περιφερείᾳ, ὅταν τὴν αὐτὴν περιφέρειαν βάσιν ἔχωσιν·
Similarly we shall prove that the angle ΗΕΓ is double of the angle ΕΔΓ, of which the angle ΗΕΒ is double of the angle ΕΔΒ; therefore the remaining angle ΒΕΓ is double of the angle ΒΔΓ. Therefore in a circle the angle at the center is double of the angle at the circumference, when they have the same circumference as base.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.
§3.prop.21ἐν κύκλῳ αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι ἀλλήλαις εἰσίν.
In a circle the angles in the same segment are equal to one another.
ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν τῷ αὐτῷ τμήματι τῷ ΒΑΕΔ γωνίαι ἔστωσαν αἱ ὑπὸ ΒΑΔ, ΒΕΔ· λέγω, ὅτι αἱ ὑπὸ ΒΑΔ, ΒΕΔ γωνίαι ἴσαι ἀλλήλαις εἰσίν.
Let ΑΒΓΔ be a circle, and let the angles in the same segment ΒΑΕΔ be ΒΑΔ, ΒΕΔ; I say that the angles ΒΑΔ, ΒΕΔ are equal to one another.
εἰλήφθω γὰρ τοῦ ΑΒΓΔ κύκλου τὸ κέντρον, καὶ ἔστω τὸ Ζ, καὶ ἐπεζεύχθωσαν αἱ ΒΖ, ΖΔ. καὶ ἐπεὶ ἡ μὲν ὑπὸ ΒΖΔ γωνία πρὸς τῷ κέντρῳ ἐστίν, ἡ δὲ ὑπὸ ΒΑΔ πρὸς τῇ περιφερείᾳ, καὶ ἔχουσι τὴν αὐτὴν περιφέρειαν βάσιν τὴν ΒΓΔ, ἡ ἄρα ὑπὸ ΒΖΔ γωνία διπλασίων ἐστὶ τῆς ὑπὸ ΒΑΔ. διὰ τὰ αὐτὰ δὴ ἡ ὑπὸ ΒΖΔ καὶ τῆς ὑπὸ ΒΕΔ ἐστι διπλασίων· ἴση ἄρα ἡ ὑπὸ ΒΑΔ τῇ ὑπὸ ΒΕΔ. ἐν κύκλῳ ἄρα αἱ ἐν τῷ αὐτῷ τμήματι γωνίαι ἴσαι ἀλλήλαις εἰσίν·
For let the center of the circle ΑΒΓΔ be taken, and let it be Ζ, and let ΒΖ, ΖΔ be joined. And since the angle ΒΖΔ is at the center, and the angle ΒΑΔ at the circumference, and they have the same circumference ΒΓΔ as base, therefore the angle ΒΖΔ is double of the angle ΒΑΔ. For the same reason the angle ΒΖΔ is also double of the angle ΒΕΔ; therefore the angle ΒΑΔ is equal to the angle ΒΕΔ. Therefore in a circle the angles in the same segment are equal to one another.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.
§3.prop.22τῶν ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.
The opposite angles of quadrilaterals in circles are equal to two right angles.
ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν αὐτῷ τετράπλευρον ἔστω τὸ ΑΒΓΔ· λέγω, ὅτι αἱ ἀπεναντίον γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.
Let ΑΒΓΔ be a circle, and let ΑΒΓΔ be a quadrilateral in it; I say that the opposite angles are equal to two right angles.
ἐπεζεύχθωσαν αἱ ΑΓ, ΒΔ. ἐπεὶ οὖν παντὸς τριγώνου αἱ τρεῖς γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν, τοῦ ΑΒΓ ἄρα τριγώνου αἱ τρεῖς γωνίαι αἱ ὑπὸ ΓΑΒ, ΑΒΓ, ΒΓΑ δυσὶν ὀρθαῖς ἴσαι εἰσίν.
Let ΑΓ, ΒΔ be joined. Since then the three angles of any triangle are equal to two right angles, therefore the three angles of the triangle ΑΒΓ, namely the angles ΓΑΒ, ΑΒΓ, ΒΓΑ, are equal to two right angles.
ἴση δὲ ἡ μὲν ὑπὸ ΓΑΒ τῇ ὑπὸ ΒΔΓ· ἐν γὰρ τῷ αὐτῷ τμήματί εἰσι τῷ ΒΑΔΓ· ἡ δὲ ὑπὸ ΑΓΒ τῇ ὑπὸ ΑΔΒ· ἐν γὰρ τῷ αὐτῷ τμήματί εἰσι τῷ ΑΔΓΒ· ὅλη ἄρα ἡ ὑπὸ ΑΔΓ ταῖς ὑπὸ ΒΑΓ, ΑΓΒ ἴση ἐστίν.
And the angle ΓΑΒ is equal to the angle ΒΔΓ, for they are in the same segment ΒΑΔΓ; and the angle ΑΓΒ to the angle ΑΔΒ, for they are in the same segment ΑΔΓΒ; therefore the whole angle ΑΔΓ is equal to the angles ΒΑΓ, ΑΓΒ.
κοινὴ προσκείσθω ἡ ὑπὸ ΑΒΓ· αἱ ἄρα ὑπὸ ΑΒΓ, ΒΑΓ, ΑΓΒ ταῖς ὑπὸ ΑΒΓ, ΑΔΓ ἴσαι εἰσίν.
Let the angle ΑΒΓ be added as common; therefore the angles ΑΒΓ, ΒΑΓ, ΑΓΒ are equal to the angles ΑΒΓ, ΑΔΓ.
ἀλλʼ αἱ ὑπὸ ΑΒΓ, ΒΑΓ, ΑΓΒ δυσὶν ὀρθαῖς ἴσαι εἰσίν. καὶ αἱ ὑπὸ ΑΒΓ, ΑΔΓ ἄρα δυσὶν ὀρθαῖς ἴσαι εἰσίν.
But the angles ΑΒΓ, ΒΑΓ, ΑΓΒ are equal to two right angles; therefore the angles ΑΒΓ, ΑΔΓ are also equal to two right angles.
ὁμοίως δὴ δείξομεν, ὅτι καὶ αἱ ὑπὸ ΒΑΔ, ΔΓΒ γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν.
Similarly we shall prove that the angles ΒΑΔ, ΔΓΒ are also equal to two right angles.
τῶν ἄρα ἐν τοῖς κύκλοις τετραπλεύρων αἱ ἀπεναντίον γωνίαι δυσὶν ὀρθαῖς ἴσαι εἰσίν·
Therefore the opposite angles of quadrilaterals in circles are equal to two right angles.
ὅπερ ἔδει δεῖξαι.
Which was to be proved.

Notes

  1. 3.prop.20Κεκλάσθω — Third-person singular perfect passive imperative of κλάω ('to bend, break'). In Greek geometry, it functions as a technical expression to introduce a different case where the lines are bent back to form another configuration (specifically, when the center of the circle lies outside the angle).
  2. 3.prop.20ὧν — Genitive plural of the relative pronoun ὅς, functioning as a partitive genitive. It indicates that from the previously mentioned whole relationship (angle ΗΕΓ being double of ΕΔΓ), a specific part (angle ΗΕΒ being double of ΕΔΒ) is being subtracted to obtain the remaining relationship.
  3. 3.prop.21ἐν τῷ αὐτῷ τμήματι — Composed of the preposition ἐν and the dative τμήματι (from τμῆμα, 'segment of a circle'). In Greek mathematical terminology, angles are described as being 'in the same segment' when their vertices lie on the arc of that segment and their sides pass through the extremities of the segment's base.
  4. 3.prop.22κοινὴ προσκείσθω — A combination of the feminine nominative singular adjective κοινός ('common') and the third-person singular present imperative of πρόσκειμαι ('to be added'). A formulaic expression indicating the algebraic/geometric operation of adding a common term to both sides of an equation.

Cite this passage

Euclid, Elements §3.prop.20-3.prop.22. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:3.prop.20-3.prop.22

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