# Fragmenta 6.34

Euclid, Fragmenta, Chapter 6. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/euclid/fragmenta/6.34

λδ΄. Ἔστω, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ ΑΔ πρὸς τὴν ∠Γ. καὶ τετμήσθω ἡ ΑΓ δίχα κατὰ τὸ Ε σημεῖον· ὅτι γίνεται τρία, τὸ μὲν ὑπὸ ΒΕ∠ ἴσον τῷ ἀπὸ ΕΓ τετραγώνῳ, τὸ δὲ ὑπὸ Β∠Ε τῷ ὑπὸ Α∠Γ, τὸ δὲ ὑπὸ ΑΒΓ τῷ ὑπὸ ΕΒ∠. ἐπεὶ γάρ, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ Α∠ πρὸς τὴν ∠Γ, συνθέντι καὶ τὰ ἡμίση τῶν ἡγουμένων καὶ ἀναστρέψαντι ἄρα ἐστίν, ὡς ἡ ΒΕ πρὸς τὴν ΕΓ, ἡ ΕΓ πρὸς τὴν Ε∠· τὸ ἄρα ὑπὸ ΒΕ∠ ἴσον ἐστὶν τῷ ἀπὸ ΕΓ. κοινὸν ἀφῃρήσθω τὸ ἀπὸ ΔΕ τετράγωνον· λοιπὸν ἄρα τὸ ὑπὸ Β∠Ε ἴσον ἐστὶν τῷ ὑπὸ Α∠Γ. πάλιν τὸ ὑπὸ ΒΕ∠ ἴσον ἐστὶν τῷ ἀπὸ ΕΓ τετραγώνῳ. ἀμφότερα ἀφῃρήσθω ἀπὸ τοῦ ἀπὸ τῆς ΒΕ τετραγώνου· λοιπὸν ἄρα τὸ ὑπὸ τῶν ΑΒΓ ἴσον ἐστὶν τῷ ὑπὸ τῶν ΕΒ∠. Ἀλλὰ ἔστω νῦν τὸ ὑπὸ τῶν Β∠Ε ἴσον τῷ ὑπὸ τῶν Α∠Γ, καὶ τετμήσθω δίχα ἡ ΓΑ κατὰ τὸ Ε· ὅτι ἐστίν, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ Α∠ πρὸς τὴν ΔΓ. ἐπεὶ γὰρ τὸ ὑπὸ τῶν Β∠Ε ἴσον ἐστὶν τῷ ὑπὸ τῶν Α∠Γ, κοινὸν προσκείσθω τὸ ἀπὸ ∠Ε τετράγωνον· ὅλον ἄρα τὸ ὑπὸ ΒΕ∠ ἴσον τῷ ἀπὸ ΓΕ τετραγώνῳ. ἀνάλογον καὶ ἀναστρέφοντι καὶ δὶς τὰ ἡγούμενα καὶ διελόντι ἄρα ἐστίν, ὡς ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ Α∠ πρὸς τὴν ∠Γ.

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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/.