# Synagoge 3.21

Pappus Alexandrinus, Synagoge, Book 3. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/3.21

21 Καὶ τὸ ἀναστρόφιον δὲ φανερόν ἐστιν. ἐὰν γὰρ ᾖ ὡς ἡ ΑΒ πρὸς ΒΓ, οὕτως ἡ ΑΕ πρὸς ΕΓ, ἡ Κέστιν ἐπʼ τὐθείας γίνεται τῇ ∠Ε. Παράλληλός τε γάρ ἐστιν ἡ ΑΚ τῇ Γ ∠ καὶ ἔστιν ὡς ἡ ΑΒ πρὸς βΓ, τουτέστιν ὡς ἡ ΑΕ πρὸς ΕΓ, ἡ ΑΚ πρὸς Γ ∠· ἐπʼ εὐθείας ἄρα ἐστὶν ἡ Κ ∠ τῇ ∠Ε. εἰ γὰρ ἡ διὰ τῶν Κ Ε οὐχ ἥξει καὶ διὰ τοῦ ∠, ἀλλὰ διὰ τοῦ Θ, γίνεται ὡς ἡ ΑΕ πρὸς ΕΓ, ἡ ΑΚ πρὸς ΓΘ, ὅπερ ἀδύνατον. ὁμοίως οὐδὲ τοῦ ∠ ἐκτὸς ἥξει τέμνουσα τὴν Γ ∠ ἐκβληθεῖσαν, βληθεἴσαν, οἷον κατὰ τὸ Ν· ἔσται γὰρ πάλιν ὡς ἡ ΑΕ πρὸς ΕΓ, ἡ ΑΚ πρὸς ΓΝ, ὅπερ ἀδύνατον ἔστιν γὰρ πρὸς τὴν Γ ∠. Ἢ οὕτως. διὰ τοῦ Κ τῇ ΑΕ παράλληλος ἡ ΚΝ ἤχθω, καὶ γίνεται παραλληλόγραμμον τὸ ΑΓΝΚ, καὶ ἴση ἡ ΑΚ τῇ ΓΝ. καὶ ἐπεί ἐστιν ὡς ἡ ΑΕ πρὸς ΕΓ, οὕτως ἡ ΑΚ, τουτέστιν ἡ ΓΝ, πρὸς Γ ∠ , διελόντι ὡς ἡ ΑΓ πρὸς ΓΕ, ἡ Ν ∠ τρὸς ∠Γ. ἐναλλὰξ ὡς ἡ ΑΓ, τουτέστιν ὡς ἡ ΚΝ, πρὸς Ν ∠, οὕτως ἡ ΕΓ πρὸς Γ ∠. καὶ περὶ τὰς ἴσας γωνίας τὰς πρὸς τοῖς Ν Γ αἱ πλευραὶ ἀνάλογόν εἰσιν ὅμοιον ἄρα ἐστὶν τὸ Ε ∠Γ τρίγωνον τῷ ∠ΝΚ τριγώνῳ· ἴση ἄρα ἐστὶν ἡ ὑπὸ Ε ∠Γ γωνία τῇ ὑπὸ Ν ∠Κ. καὶ ἔστιν εὐθεῖα ἡ ΓΝ· εὐθεῖα ἄρα καὶ ἡ Κ ∠Ε.

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Pappus Alexandrinus, Synagoge, ed. Friedrich Hultsch. 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/tlg2032.tlg001.local001/.