§6.1α΄.
1.
Ἔστω καταγραφὴ ἡ ΑΒΓ∠ΕΖΗ, καὶ ἔστω, ὡς ἡ ΑΖ πρὸς τὴν ΖΗ, οὕτως ἡ Α∠ πρὸς τὴν ∠Γ, καὶ ἐπεζεύχθω ἡ ΘΚ ὅτι παράλληλός ἐστιν ἡ ΘΚ τῇ ΑΓ.
ἤχθω διὰ τοῦ Ζ τῇ Β∠ παράλληλος ἡ ΖΛ.
ἐπεὶ οὖν ἔστιν, ὡς ἡ ΑΖ πρὸς τὴν ΖΗ, οὕτως ἡ Α∠ πρὸς τὴν ∠Γ ἀνάπαλιν καὶ συνθέντι καὶ ἐναλλάξ ἐστιν, ὡς ἡ ∠Α πρὸς τὴν ΑΖ, τουτέστιν ἐν παραλλήλῳ ὡς ἡ ΒΑ πρὸς τὴν ΑΛ, οὕτως ἡ ΓΑ πρὸς τὴν ΑΗ παράλληλος ἄρα ἐστὶν ἡ ΛΗ τῇ ΒΓ ἔστιν ἄρα, ὡς ἡ ΕΒ πρὸς τὴν ΒΛ, οὕτως ἐν παραλλήλῳ ἡ ΕΘ πρὸς τὴν ΘΗ. ἔστι δὲ καί, ὡς ἡ ΕΒ πρὸς τὴν ΒΛ, οὕτως ἐν παραλλήλῳ ἡ ΕΚ πρὸς τὴν ΚΖ καὶ ὡς ἄρα ἡ ΕΚ πρὸς τὴν ΚΖ, οὕτως ἐστὶν ἡ ΕΘ πρὸς τὴν ΘΗ. παράλληλος ἄρα ἐστὶν ἡ ΘΚ τῇ ΑΓ.
Διὰ δὲ τοῦ συνημμένου οὕτως· ἐπεί ἐστιν, ὡς ἡ ΑΖ πρὸς τὴν ΖΗ, οὕτως ἡ Α∠ πρὸς τὴν ∠Γ. ἀνάπαλίν ἐστιν, ὡς ἡ ΗΖ πρὸς τὴν ΖΑ, οὕτως ἡ Γ∠ πρὸς τὴν ∠Α. συνθέντι καὶ ἐναλλὰξ καὶ ἀναστρέψαντί ἐστιν, ὡς ἡ Α∠ πρὸς τὴν ∠Ζ, οὕτως ἡ ΑΓ πρὸς τὴν ΓΗ. ἀλλʼ ὁ μὲν τῆς Α∠ πρὸς τὴν ∠Ζ συνῆπται ἔκ τε τοῦ τῆς ΑΒ πρὸς τὴν ΒΕ καὶ τοῦ τῆς ΕΚ πρὸς τὴν ΚΖ, ὁ δὲ τῆς ΑΓ πρὸς τὴν ΓΗ ἔκ τε τοῦ τῆς Α Β πρὸς τὴν ΒΕ καὶ τοῦ τῆς ΕΘ πρὸς τὴν ΘΗ ὁ ἄρα συνημμένος λόγος ἔκ τε τοῦ, ὃν ἔχει ἡ ΑΒ πρὸς τὴν ΒΕ, καὶ ἡ ΕΚ πρὸς τὴν ΚΖ, ὁ αὐτός ἔστιν τῷ συνημμένῳ ἔκ τε τοῦ, ὃν ἔχει ἡ ΑΒ πρὸς τὴν ΒΕ, καὶ ἡ ΕΘ πρὸς τὴν ΘΗ. καὶ κοινὸς ἐκκεκρούσθω ὁ τῆς ΑΒ πρὸς τὴν ΒΕ λόγος· λοιπὸν ἄρα ὁ τῆς ΕΚ πρὸς τὴν ΚΖ λόγος ὁ αὐτός ἐστιν τῷ τῆς ΕΘ πρὸς τὴν ΘΗ. παράλληλος ἄρα ἐστὶν ἡ ΘΚ τῇ ΑΓ.
Let there be the diagram ΑΒΓ∠ΕΖΗ, and let it be that, as ΑΖ is to ΖΗ, so is Α∠ to ∠Γ, and let ΘΚ be joined; (to prove) that ΘΚ is parallel to ΑΓ. Let ΖΛ be drawn through Ζ parallel to Β∠. Since therefore, as ΑΖ is to ΖΗ, so is Α∠ to ∠Γ, by inversion and composition and alternation, as ∠Α is to ΑΖ, that is, in the parallel, as ΒΑ is to ΑΛ, so is ΓΑ to ΑΗ; therefore ΛΗ is parallel to ΒΓ. Therefore, as ΕΒ is to ΒΛ, so, in the parallel, is ΕΘ to ΘΗ. But also, as ΕΒ is to ΒΛ, so, in the parallel, is ΕΚ to ΚΖ; and therefore as ΕΚ is to ΚΖ, so is ΕΘ to ΘΗ. Therefore ΘΚ is parallel to ΑΓ. But by the compounded ratio thus: Since as ΑΖ is to ΖΗ, so is Α∠ to ∠Γ, by inversion as ΗΖ is to ΖΑ, so is Γ∠ to ∠Α. By composition and alternation and conversion, as Α∠ is to ∠Ζ, so is ΑΓ to ΓΗ. But the ratio of Α∠ to ∠Ζ is compounded of that of ΑΒ to ΒΕ and that of ΕΚ to ΚΖ, while the ratio of ΑΓ to ΓΗ is compounded of that of ΑΒ to ΒΕ and that of ΕΘ to ΘΗ. Therefore the compounded ratio of that which ΑΒ has to ΒΕ and ΕΚ to ΚΖ is the same as the compounded ratio of that which ΑΒ has to ΒΕ and ΕΘ to ΘΗ. And let the common ratio of ΑΒ to ΒΕ be removed; therefore the remaining ratio of ΕΚ to ΚΖ is the same as that of ΕΘ to ΘΗ. Therefore ΘΚ is parallel to ΑΓ.
§6.2## Εἰς τὸ δεύτερον πόρισμα. β΄.
## For the second porism. 2.
Καταγραφὴ ἡ ΑΒΓ∠ΕΖΗΘ, ἔστω δὲ παράλληλος ἡ ΑΖ τῇ ∠Β, ὡς δὲ ἡ ΑΕ πρὸς τὴν ΕΖ, οὕτως ἡ ΓΗ πρὸς τὴν ΗΖ ὅτι εὐθεῖά ἐστιν ἡ διὰ τῶν Θ, Κ, Ζ.
ἤχθω διὰ τοῦ Η παρὰ τὴν ∠Ε ἡ ΗΛ, καὶ ἐπιζευχθεῖσα ἡ ΘΚ ἐκβεβλήσθω ἐπὶ τὸ Λ.
ἐπεῖ οὖν ἐστιν, ὡς ἡ ΑΕ πρὸς τὴν ΕΖ, οὕτως ἡ ΓΗ πρὸς τὴν ΗΖ, ἐναλλάξ ἐστιν, ὡς ἡ ΑΕ πρὸς τὴν ΓΗ οὕτως ἡ ΕΖ πρὸς τὴν ΖΗ. ὡς δὲ ἡ ΑΕ πρὸς τὴν ΓΗ οὕτως ἡ ΕΘ πρὸς τὴν ΗΛ καὶ ὡς ἀρα ἡ Ε πρὸς τὴν ΖΕ, οὕτως ἡ ΕΘ πρὸς τὴν ΗΛ. καί ἔστι παράλληλος ἡ ΕΘ τῇ ΗΛ· εὐθεῖα ἄρα ἐστὶν ἡ διὰ τῶν Θ, Λ, Ζ· ὅπερ ἔδει δεῖξαι.
Let the diagram be ΑΒΓ∠ΕΖΗΘ, and let ΑΖ be parallel to ∠Β, and as ΑΕ is to ΕΖ, so is ΓΗ to ΗΖ; (to prove) that the line through Θ, Κ, Ζ is a straight line. Let ΗΛ be drawn through Η parallel to ∠Ε, and let ΘΚ be joined and produced to Λ. Since therefore, as ΑΕ is to ΕΖ, so is ΓΗ to ΗΖ, alternately, as ΑΕ is to ΓΗ, so is ΕΖ to ΖΗ. But as ΑΕ is to ΓΗ, so is ΕΘ to ΗΛ; and therefore as ΕΖ is to ΖΗ, so is ΕΘ to ΗΛ. And ΕΘ is parallel to ΗΛ; therefore the line through Θ, Λ, Ζ is a straight line; which was to be proved.