Chapter 6

ιδʹ. Ἔστω παράλληλος ἡ ΑΒ τῇ Γ∠, καὶ διήχθωσαν αἱ ΑΕ, ΓΒ, καὶ σημεῖον ἐπὶ τῆς ΒΗ τὸ Ζ, ὥστε εἶναι, ὡς τὴν ∠Ε πρὸς τὴν ΕΓ, οὕτως τὸ ὑπὸ ΓΒ, ΗΖ πρὸς τὸ ὑπὸ ΖΒ, ΓΗ ὅτι εὐθεῖά ἐστιν ἡ διὰ τῶν Α, Ζ, ∠. ἤχθω διὰ μὲν τοῦ ∠τῇ ΒΓ παράλληλος ἡ ∠Θ, καὶ ἐκβεβλήσθω ἡ ΑΕ ἐπὶ τὸ Θ, διὰ δὲ τοῦ Θ τῇ Γ∠ παράλληλος ἡ Θ Κ, καὶ ἐκβεβλήσθω ἡ Β ἐπὶ τὸ Κ. ἐπεὶ οὖν ἐστιν, ὡς ἡ ∠Ε πρὸς τὴν ΕΓ. οὕτως τὸ ὑπὸ ΓΒ, Ζ πρὸς τὸ ὑπὸ ΒΖ, ΓΗ, ὡς δὲ ἡ ∠Ε πρὸς τὴν ΕΓ, οὕτως ἐστὶν ἥ τε ∠Θ πρὸς τὴν Γ καὶ τὸ ὑπὸ ∠Θ, ΒΖ πρὸς τὸ ὑπὸ τῶν ΓΗ, ἴσον ἄρα ἐστὶν τὸ ὑπὸ τῶν ΒΓ ΖΗ τῷ ὑπὸ ∠Θ, ΒΖ ἀνάλογον ἄρα ἐστίν, ὡς ἡ ΓΒ πρὸς τήν ΒΖ, οὕτως ἡ ∠Θ, τουτέστιν ἡ Γ Κ, πρὸς τὴν ΗΖ καὶ ὅλη ἄρα ἡ ΚΒ πρὸς ὅλην τὴν Β ἐστιν, ὡς ἡ ΚΓ πρὸς ΖΗ τουτέστιν ὡς ἡ ∠Θ πρὸς ΖΗ. ἀλλʼ ὡς ἡ ΚΒ πρὸς Β ἐν παραλλήλῳ, οὕτως ἐστὶν ἡ ΘΑ πρὸς ΑΗ καὶ ἡ ∠Θ πρὸς ΖΗ. καί εἰσιν παράλληλοι αἱ ∠Θ, ΖΗ· εὐθεῖα ἄρα ἐστὶν ἡ διὰ τῶν Α, Ζ, ∠σημείων.