Section 18

Ἐὰν δύο εὐθεῖαι δοθὲν χωρίον περιέχωσιν ἐν δεδο- μένῃ γωνίᾳ, τὸ δὲ ἀπὸ τῆς μείζονος τοῦ ἀπὸ τῆς ἐλάσσονος δοθέντι μεῖζον ᾖ, καὶ ἑκατέρα αὐτῶν ἔσται δοθεῖσα. δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΒΓ δοθὲν περιεχέτωσαν χωρίον τὸ ΑΓ ἐν δεδομένῃ γωνίᾳ τῇ ὑπὸ τῶν ΑΒΓ, τὸ δὲ ἀπὸ τῆς ΑΒ δοθέντι μεῖζον ἔστω τοῦ ἀπὸ τῆς ΒΓ· λέγω, ὅτι δοθεῖσά ἐστιν ἑκατέρα τῶν ΑΒ, ΒΓ. ἐπεὶ γὰρ τὸ ἀπὸ τῆς ΑΒ τοῦ ἀπὸ τῆς ΒΓ δοθέντι μεῖζόν ἐστιν, ἀφῃρήσθω τὸ δοθὲν τὸ ὑπὸ τῶν ΑΒ, ΒΔ· λοιπὸν ἄρα τὸ ὑπὸ τῶν BA, ΑΔ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΓ. καὶ ἐπεὶ δοθέν ἐστι τὸ ὑπὸ τῶν ΑΒ, ΒΓ, ἔστι δὲ καὶ τὸ ὑπὸ τῶν ΑΒ, ΒΔ δοθέν, λόγος ἄρα τοῦ ὑπὸ τῶν ΑΒ, ΒΔ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΓ δοθείς. καί ἐστιν, ὡς τὸ ὑπὸ τῶν ΑΒ, ΒΔ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΓ, οὕτως ἡ ΔΒ πρὸς ΒΓ· λόγος ἄρα καὶ τῆς ΔΒ πρὸς ΒΓ δοθείς· λόγος ἄρα καὶ τοῦ ἀπὸ τῆς ΔΒ πρὸς τὸ ἀπὸ τῆς ΒΓ δοθείς. τῷ δὲ ἀπὸ τῆς ΓΒ ἴσον τὸ ὑπὸ τῶν ΒΑ, ΑΔ· λόγος ἄρα καὶ τοῦ ὑπὸ τῶν ΒΑ, ΑΔ πρὸς τὸ ἀπὸ τῆς ΔΒ δοθείς· καὶ τοῦ τετράκις ἄρα ὑπὸ τῶν ΒΑ, ΑΔ μετὰ τοῦ ἀπὸ τῆς ΔΒ πρὸς τὸ ἀπὸ τῆς ΒΔ λόγος δοθείς. ἀλλὰ τὸ τετράκις ὑπὸ τῶν ΒΑ, ΑΔ μετὰ τοῦ ἀπὸ τῆς ΒΔ τὸ ἀπὸ συναμφοτέρου τῆς ΒΑ, ΑΔ ἐστιν· λόγος ἄρα καὶ τοῦ ἀπὸ συναμφοτέρου τῆς ΒΑ, ΑΔ πρὸς τὸ ἀπὸ τῆς ΔΒ δοθείς· λόγος ἄρα καὶ συναμφοτέρου τῆς ΒΑ, ΑΔ πρὸς ΔΒ δοθείς. καὶ συνθέντι συναμφοτέρου τῆς ΒΑ, ΑΔ μετὰ τῆς ΔΒ, τουτέστι δύο τῶν ΑΒ πρὸς ΒΔ λόγος ἐστὶ δοθείς· καὶ τῆς ΑΒ ἄρα πρὸς ΒΔ λόγος ἐστὶ δοθείς. τῆς δὲ ΔΒ πρὸς τὴν ΒΓ λόγος ἐστὶ δοθείς· καὶ τῆς ΑΒ ἄρα πρὸς ΒΓ λόγος δοθείς. καὶ ἐπεὶ λόγος τῆς ΑΒ πρὸς ΒΔ δοθείς, καί ἐστιν, ὡς ἡ ΑΒ πρὸς ΒΔ, οὕτως τὸ ἀπὸ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΔ, λόγος ἄρα καὶ τοῦ ἀπὸ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΑΒ, ΒΔ δοθείς. δοθὲν δὲ τὸ ὑπὸ τῶν ΑΒ, ΒΔ· οὕτως γὰρ δοθὲν ἀφῄρηται· δοθὲν ἄρα καὶ τὸ ἀπὸ τῆς ΑΒ· δοθεῖσα ἄρα ἡ ΑΒ. καί ἐστι λόγος τῆς ΑΒ πρὸς ΒΓ δοθείς· δοθεῖσα ἄρα καὶ ἡ ΒΓ.