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Euclid · Elements §3.prop.14

Equal Chords and Distance from the Center of a Circle

Passage 43 of 316 · Greek

Summary

Proves, using the Pythagorean theorem, that equal chords in a circle are equidistant from the center, and conversely, chords equidistant from the center are equal in length.

§3.prop.14ἐν κύκλῳ αἱ ἴσαι εὐθεῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι ἀλλήλαις εἰσίν.
In a circle equal straight lines are equally distant from the center, and those which are equally distant from the center are equal to one another.
ἔστω κύκλος ὁ ΑΒΓΔ, καὶ ἐν αὐτῷ ἴσαι εὐθεῖαι ἔστωσαν αἱ ΑΒ, ΓΔ· λέγω, ὅτι αἱ ΑΒ, ΓΔ ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου.
Let ΑΒΓΔ be a circle, and in it let ΑΒ, ΓΔ be equal straight lines; I say that ΑΒ, ΓΔ are equally distant from the center.
εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓΔ κύκλου καὶ ἔστω τὸ Ε, καὶ ἀπὸ τοῦ Ε ἐπὶ τὰς ΑΒ, ΓΔ κάθετοι ἤχθωσαν αἱ ΕΖ, ΕΗ, καὶ ἐπεζεύχθωσαν αἱ ΑΕ, ΕΓ. ἐπεὶ οὖν εὐθεῖά τις διὰ τοῦ κέντρου ἡ ΕΖ εὐθεῖάν τινα μὴ διὰ τοῦ κέντρου τὴν ΑΒ πρὸς ὀρθὰς τέμνει, καὶ δίχα αὐτὴν τέμνει.
For let the center of the circle ΑΒΓΔ be taken, and let it be Ε, and from Ε let ΕΖ, ΕΗ be drawn perpendicular to ΑΒ, ΓΔ, and let ΑΕ, ΕΓ be joined. Since therefore a straight line ΕΖ through the center cuts a straight line ΑΒ not through the center at right angles, it also cuts it in half.
ἴση ἄρα ἡ ΑΖ τῇ ΖΒ· διπλῆ ἄρα ἡ ΑΒ τῆς ΑΖ. διὰ τὰ αὐτὰ δὴ καὶ ἡ ΓΔ τῆς ΓΗ ἐστι διπλῆ· καί ἐστιν ἴση ἡ ΑΒ τῇ ΓΔ·
Therefore ΑΖ is equal to ΖΒ; therefore ΑΒ is double of ΑΖ. For the same reason indeed ΓΔ is also double of ΓΗ; and ΑΒ is equal to ΓΔ; therefore ΑΖ is also equal to ΓΗ.
ἴση ἄρα καὶ ἡ ΑΖ τῇ ΓΗ. καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΕ τῇ ΕΓ, ἴσον καὶ τὸ ἀπὸ τῆς ΑΕ τῷ ἀπὸ τῆς ΕΓ. ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΕ ἴσα τὰ ἀπὸ τῶν ΑΖ, ΕΖ· ὀρθὴ γὰρ ἡ πρὸς τῷ Ζ γωνία· τῷ δὲ ἀπὸ τῆς ΕΓ ἴσα τὰ ἀπὸ τῶν ΕΗ, ΗΓ· ὀρθὴ γὰρ ἡ πρὸς τῷ Η γωνία· τὰ ἄρα ἀπὸ τῶν ΑΖ, ΖΕ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΓΗ, ΗΕ, ὧν τὸ ἀπὸ τῆς ΑΖ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΗ· ἴση γάρ ἐστιν ἡ ΑΖ τῇ ΓΗ· λοιπὸν ἄρα τὸ ἀπὸ τῆς ΖΕ τῷ ἀπὸ τῆς ΕΗ ἴσον ἐστίν· ἴση ἄρα ἡ ΕΖ τῇ ΕΗ. ἐν δὲ κύκλῳ ἴσον ἀπέχειν ἀπὸ τοῦ κέντρου εὐθεῖαι λέγονται, ὅταν αἱ ἀπὸ τοῦ κέντρου ἐπʼ αὐτὰς κάθετοι ἀγόμεναι ἴσαι ὦσιν· αἱ ἄρα ΑΒ, ΓΔ ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου.
And since ΑΕ is equal to ΕΓ, the square on ΑΕ is also equal to the square on ΕΓ. But the squares on ΑΖ, ΕΖ are equal to the square on ΑΕ; for the angle at Ζ is right; and the squares on ΕΗ, ΗΓ are equal to the square on ΕΓ; for the angle at Η is right; therefore the squares on ΑΖ, ΖΕ are equal to the squares on ΓΗ, ΗΕ, of which the square on ΑΖ is equal to the square on ΓΗ; for ΑΖ is equal to ΓΗ; therefore the remainder, the square on ΖΕ, is equal to the square on ΕΗ; therefore ΕΖ is equal to ΕΗ. But in a circle straight lines are said to be equally distant from the center when the perpendiculars drawn from the center to them are equal; therefore ΑΒ, ΓΔ are equally distant from the center.
ἀλλὰ δὴ αἱ ΑΒ, ΓΔ εὐθεῖαι ἴσον ἀπεχέτωσαν ἀπὸ τοῦ κέντρου, τουτέστιν ἴση ἔστω ἡ ΕΖ τῇ ΕΗ. λέγω, ὅτι ἴση ἐστὶ καὶ ἡ ΑΒ τῇ ΓΔ. τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, ὅτι διπλῆ ἐστιν ἡ μὲν ΑΒ τῆς ΑΖ, ἡ δὲ ΓΔ τῆς ΓΗ· καὶ ἐπεὶ ἴση ἐστὶν ἡ ΑΕ τῇ ΓΕ, ἴσον ἐστὶ τὸ ἀπὸ τῆς ΑΕ τῷ ἀπὸ τῆς ΓΕ· ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΕ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΕΖ, ΖΑ, τῷ δὲ ἀπὸ τῆς ΓΕ ἴσα τὰ ἀπὸ τῶν ΕΗ, ΗΓ. τὰ ἄρα ἀπὸ τῶν ΕΖ, ΖΑ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΕΗ, ΗΓ·
But indeed let the straight lines ΑΒ, ΓΔ be equally distant from the center, that is, let ΕΖ be equal to ΕΗ. I say that ΑΒ is also equal to ΓΔ. For with the same construction we shall show in like manner that ΑΒ is double of ΑΖ, and ΓΔ of ΓΗ; and since ΑΕ is equal to ΓΕ, the square on ΑΕ is equal to the square on ΓΕ; but the squares on ΕΖ, ΖΑ are equal to the square on ΑΕ, and the squares on ΕΗ, ΗΓ are equal to the square on ΓΕ.
ὧν τὸ ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς ΕΗ ἐστιν ἴσον· ἴση γὰρ ἡ ΕΖ τῇ ΕΗ· λοιπὸν ἄρα τὸ ἀπὸ τῆς ΑΖ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΗ· ἴση ἄρα ἡ ΑΖ τῇ ΓΗ· καί ἐστι τῆς μὲν ΑΖ διπλῆ ἡ ΑΒ, τῆς δὲ ΓΗ διπλῆ ἡ ΓΔ· ἴση ἄρα ἡ ΑΒ τῇ ΓΔ. ἐν κύκλῳ ἄρα αἱ ἴσαι εὐθεῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι ἀλλήλαις εἰσίν· ὅπερ ἔδει δεῖξαι.
Therefore the squares on ΕΖ, ΖΑ are equal to the squares on ΕΗ, ΗΓ; of which the square on ΕΖ is equal to the square on ΕΗ; for ΕΖ is equal to ΕΗ; therefore the remainder, the square on ΑΖ, is equal to the square on ΓΗ; therefore ΑΖ is equal to ΓΗ; and ΑΒ is double of ΑΖ, and ΓΔ double of ΓΗ; therefore ΑΒ is equal to ΓΔ. Therefore in a circle equal straight lines are equally distant from the center, and those which are equally distant from the center are equal to one another; which was to be proved.

Notes

  1. §3.prop.14τὸ ἀπὸ τῆς ΑΕ — This refers to "the square on AE". In Greek mathematical texts, τὸ ἀπὸ ... (or τὸ ὑπὸ ...) with a neuter singular article represents the square on that line segment (or the rectangle contained by two segments).
  2. §3.prop.14ὧν τὸ ἀπὸ τῆς ΑΖ ἴσον ἐστὶ — The genitive relative pronoun ὧν refers back to the sums of squares just mentioned (τὰ ἀπὸ τῶν ΑΖ, ΖΕ and τοῖς ἀπὸ τῶν ΓΗ, ΗΕ), expressing "of which (equals)".
  3. §3.prop.14τῶν γὰρ αὐτῶν κατασκευασθέντων — A genitive absolute construction, meaning "for with the same construction being made" or "assuming the same construction".

Cite this passage

Euclid, Elements §3.prop.14. Humanitext Reader, https://reader.humanitext.ai/en/text/urn:cts:greekLit:tlg1799.tlg001.humanitext-grc2:3.prop.14

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