# Fragmenta 6.5

Euclid, Fragmenta, Chapter 6. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/euclid/fragmenta/6.5

ε΄. Ἐὰν καταγραφὴ ἡ ΑΒΓ∠ΕΖΗΘ, γίνεται, ὡς ἡ Α∠ πρὸς τὴν ∠Γ, οὕτως ἡ ΑΒ πρὸς τὴν ΒΓ. ἔστω οὖν, ὡς ἡ Α∠ πρὸς τὴν ∠Γ. οὕτως ἡ Α πρὸς τὴν ΒΓ· ὅτι εὐθεῖά ἐστιν ἡ διὰ τῶν Α, Η, Θ. ἤχθω διὰ τοῦ Η τῇ Α∠ παράλληλος ἡ ΚΛ. ἐπεὶ οὖν ἐστιν, ὡς ἡ Α∠ πρὸς τὴν ∠Γ, οὕτως ἡ ΑΒ πρὸς τὴν ΒΓ, ἀλλ᾿ ὡς μὲν ἡ Α∠ πρὸς τὴν ∠Γ οὕτως ἡ ΚΛ πρὸς τὴν ΛΗ, ὡς δὲ ἡ ΑΒ πρὸς τὴν ΒΓ, οὕτως ἡ ΚΗ πρὸς τὴν ΗΜ, καὶ ὡς ἄρα ἡ ΚΛ πρὸς τὴν ΛΗ, οὕτως ἡ ΚΗ πρὸς τὴν ΗΜ, καὶ λοιπὴ ἡ ΗΛ πρὸς λοιπὴν τὴν ΛΜ ἐστιν, ὡς ἡ ΚΛ πρὸς τὴν ΛΗ, τουτέστιν ὡς ἡ Α∠ πρὸς τὴν ∠Γ. ἐναλλάξ ἐστιν, ὡς ἡ Α∠ πρὸς τὴν ΗΛ, οὕτως ἡ Γ∠ πρὸς τὴν ΛΜ, τουτέστιν ἡ ∠Θ πρὸς ΘΛ. καί ἐστι παράλληλος ἡ ΗΛ τῇ Α∠· εὐθεῖα ἄρα ἐστὶν ἡ διὰ τῶν Α, Η, Θ σημείων· τοῦτο γὰρ φανερόν.

---

Euclid, Fragmenta, ed. Heinrich Menge. Text: The Greek Library, from the Perseus Digital Library and other open sources, CC BY-SA 4.0. Changes: apostrophes normalised and the text divided into reading sections by Interlinea.
License: CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Source: https://thegreeklibrary.pages.lisn.upsaclay.fr/works/tlg1799.tlg016.local001/.