Chapter 6

θ΄. Ἔστω τρίγωνον τὸ ΑΒΓ καὶ ἐν αὐτῷ διήχθωσαν αἱ Α∠, ΑΕ, καὶ τῇ ΒΓ παράλληλος ἤχθω ἡ ΖΗ, καὶ κεκλάσθω ἡ ΖΘΗ, ἔστω δέ, ὡς ἡ ΒΘ πρὸς τὴν ΘΓ, οὕτως ἡ ∠Θ πρὸς τὴν ΘΕ ὅτι παράλληλός ἔστιν ἡ ΚΛ τῇ ΒΓ. ἐπεὶ γάρ ἐστιν, ὡς ἡ ΒΘ πρὸς τὴν ΘΓ, οὕτως ἡ ∠Θ πρὸς τὴν ΘΕ, λοιπὴ ἄρα ἡ Β∠ πρὸς λοιπὴν τὴν ΓΕ ἐστιν, ὡς ἡ ∠Θ πρὸς τὴν ΘΕ. ὡς δὲ ἡ Β∠ πρὸς τὴν ΕΓ, οὕτως ἐστὶν ἡ ΖΜ πρὸς Ν καὶ ὡς ἄρα ἡ ΖΜ πρὸς ΝΗ, οὕτως ἐστὶν ἡ ∠Θ πρὸς τὴν ΘΕ. ἐναλλάξ ἔστιν, ὡς ἡ ΖΜ πρὸς τὴν ∠Θ, οὕτως ἡ ΝΗ πρὸς τὴν ΘΕ. ἀλλʼ ὡς μὲν ἡ ΖΜ πρὸς τὴν ∠Θ, οὕτως ἐστὶν ἐν παραλλήλῳ ἡ ΖΚ πρὸς τὴν ΚΘ, ὡς δὲ ἡ ΗΝ πρὸς τὴν ΘΕ, οὕτως ἐστὶν ἡ ΗΛ πρὸς τὴν ∠Θ καὶ ὡς ἄρα ἡ ΖΚ πρὸς τὴν Κ Θ, οὕτως ἐστὶν ἡ ΗΛ πρὸς τὴν ∠Θ. παράλληλος ἄρα ἐστὶν ἡ ΚΛ τῇ ΗΖ· ὥστε καὶ τῇ ΓΒ.