Fragmenta

δ΄. Καταγραφὴ ἡ ΑΒΓ∠ΕΖΗΘΚΛ, ἔστω δέ, ὡς τὸ ὑπὸ ΑΖ, ΒΓ πρὸς τὸ ὑπὸ ΑΒ, ΓΖ, οὕτως τὸ ὑπὸ ΑΖ, ∠Ε πρὸς τὸ ὑπὸ Α∠, ΕΖ· ὅτι εὐθεῖά ἐστιν ἡ διὰ τῶν Θ, Η, Ζ σημείων.

ἐπεί ἐστιν, ὡς τὸ ὑπὸ ΑΖ, ΒΓ πρὸς τὸ ὑπὸ ΑΒ, ΓΖ, οὕτως τὸ ὑπὸ ΑΖ, ∠Ε πρὸς τὸ ὑπὸ Α∠, ΕΖ, ἐναλλάξ ἐστιν, ὡς τὸ ὑπὸ ΑΖ, ΒΓ πρὸς τὸ ὑπὸ ΑΖ, ∠Ε, τουτέστιν ὡς ἡ ΒΓ πρὸς τὴν ∠Ε, οὕτως τὸ ὑπὸ ΑΒ, ΓΖ πρὸς τὸ ὑπὸ Α∠, ΕΖ. ἀλλ ὁ μὲν τῆς ΒΓ πρὸς τὴν ∠Ε συνῆπται λόγος, ἐὰν διὰ τοῦ Κ τῇ ΑΖ παράλληλος ἀχθῇ ἡ ΚΜ, ἔκ τε τοῦ τῆς ΒΓ πρὸς ΚΝ καὶ τῆς ΚΝ πρὸς ΚΜ καὶ ἔτι τοῦ τῆς ΚΜ πρὸς ∠Ε, ὁ δὲ τοῦ ὑπὸ ΑΒ, ΓΖ πρὸς τὸ ὑπὸ Α∠, ΕΖ συνῆπται ἔκ τε τοῦ τῆς ΒΑ πρὸς Α∠ καὶ τοῦ τῆς ΓΖ πρὸς τὴν ΖΕ. κοινὸς ἐκκεκρούσθω ὁ τῆς ΒΑ πρὸς Α∠ ὁ αὐτὸς ὢν τῷ τῆς ΝΚ πρὸς ΚΜ· λοιπὸν ἄρα ὁ τῆς ΓΖ πρὸς τὴν ΖΕ συνῆπται ἔκ τε τοῦ τῆς ΒΓ πρὸς τὴν ΚΝ, τουτέστιν τοῦ τῆς ΘΓ πρὸς τὴν ΚΘ, καὶ τοῦ τῆς ΚΜ πρὸς τὴν ∠Ε, τουτέστιν τοῦ τῆς ΚΗ πρὸς τὴν ΗΕ. εὐθεῖα ἄρα ἡ διὰ τῶν Θ, Η, Ζ.

ἐὰν γὰρ διὰ τοῦ Ε τῇ Θ παράλληλον ἀγάγω τὴν ΕΞ, καὶ ἐπιζευχθεῖσα ἡ ΘΗ ἐκβληθῇ ἐπὶ τὸ Ξ, ὁ μὲν τῆς ΚΗ πρὸς τὴν ΗΕ λόγος ὁ αὐτός ἐστιν τῷ τῆς ΚΘ πρὸς τὴν ΕΞ, ὁ δὲ συνημμένος ἔκ τε τοῦ τῆς Γ Θ πρὸς τὴν ΘΚ καὶ τοῦ τῆς ΘΚ πρὸς τὴν ΕΞ μεταβαλλόμενος εἰς τὸν τῆς ΘΓ πρὸς ΕΞ λόγον, καὶ ὁ τῆς Γ Ζ πρὸς ΖΕ λόγος ὁ αὐτὸς τῷ τῆς ΓΘ πρὸς τὴν ΕΞ παραλλήλου οὔσης τῆς ΓΘ τῇ ΕΞ εὐθεῖα ἄρα ἐστὶν ἡ διὰ τῶν Θ, Ξ Ζ· τοῦτο γὰρ φανερόν· ὥστε καὶ ἡ διὰ τῶν Θ, Η, Ζ εὐθεῖά ἐστιν.