# Fragmenta 6.31

Euclid, Fragmenta, Chapter 6. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/euclid/fragmenta/6.31

λα΄. Ἡμικύκλιον τὸ ἐπὶ τῆς ΑΒ, καὶ ἀπὸ τῶν Α,Β σημείων τῇ ΑΒ πρὸς ὀρθὰς γωνίας εὐθεῖαι γραμμαὶ ἤχθωσαν αἱ ΒΔ, ΑΕ, καὶ ἤχθω τυχοῦσα ἡ ΔΕ, καὶ ἀπὸ τοῦ τῇ ∠Ε πρὸς ὀρθὰς γωνίας εὐθεῖα γραμμὴ ἡ ΖΗ συμπιπτέτω τῇ ΑΒ κατὰ τὸ Η ὅτι τὸ ὑπὸ τῶν ΑΕ, ΒΔ ἴσον ἐστὶν τῷ ὑπὸ τῶν ΑΗΒ. ὅτι ἄρα ἐστίν, ὡς ἡ ΕΑ πρὸς τὴν ΑΗ, οὕτως ἡ ΗΒ πρὸς τὴν Β∠. περὶ ἴσας γωνίας ἀνάλογόν εἰσιν αἱ πλευραί· ὅτι ἄρα ἴση ἐστὶν ἡ ὑπὸ τῶν ΑΗΕ γωνία τῇ ὑπὸ τῶν ΒΔΗ γωνίᾳ. ἀλλὰ ἡ μὲν ὑπὸ ΑΗΕ ἴση ἐστὶν ἐν τῷ αὐτῷ τμήματι τῇ ὑπὸ ΑΖΕ, ἡ δὲ ὑπὸ Β∠Η πάλιν ἐν τῷ αὐτῷ τμήματι τῇ ὑπὸ ΒΖΗ ὅτι ἄρα ἴση ἐστὶν ἡ ὑπὸ ΑΖΕ γωνία τῇ ὑπὸ ΒΖΗ γωνίᾳ. ἔστιν δέ· ὀρθὴ γάρ ἐστιν ἑκατέρα τῶν ὑπὸ ΑΖΒ, ΕΖΗ γωνιῶν.

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Euclid, Fragmenta, ed. Heinrich Menge. Text: The Greek Library, from the Perseus Digital Library and other open sources, CC BY-SA 4.0. Changes: apostrophes normalised and the text divided into reading sections by Interlinea.
License: CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Source: https://thegreeklibrary.pages.lisn.upsaclay.fr/works/tlg1799.tlg016.local001/.