Chapter 6

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.