# Synagoge 3.17

Pappus Alexandrinus, Synagoge, Book 3. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/3.17

17 τγ΄. Ἔστω ἡμικύκλιον τὸ ΑΒΓ, καὶ κεκλάσθω ἡ ΑΒ ∠, καὶ ἔστω ἴση ἡ ΑΒ τῇ Β ∠, καὶ τῇ Β ∠ πρὸς ὀρθάς ἤχθω ἡ ∠Ε, καὶ ἐπεζεύχθω ἡ ΒE, καὶ αὐτῇ πρὺς ὀρθὰς ἤχθω ἡ ΕΖ, καὶ τὸ κέντρον τὸ Η, καὶ ἔστω ὡς ἡ ΑΗ πρὸς Η ∠, οὕτως ἡ ∠Θ πρὸς ΘΖ, καὶ ἐπεζεύχθω ἡ ΘΕ· ὅτι ἡ ὑπὸ τῶν ΒΕ ∠ γωνία ἴση ἐστὶν τῇ ὑπὸ τῶν ∠ΕΘ γωνίᾳ. Ἤχθω ἀπὸ τοῦ ἐπὶ τὴν ΒΕ κάθετος ἡ ΗΚ· ἄρα ἐστὶν ἡ ΒΚ τῇ ΚΕ. καὶ ἔστιν ἀρθὴ ἡ ὑπὸ τῶν Β ∠E· αἱ τρεῖς ἄρα αἱ BΚ Κ ∠ ΚΕ ἴσαι ἀλλήλαις εἰσίν. καὶ παράλληλός ἐστιν ἡ ΗK τῇ ΕΖ, καὶ ἐπεὶ ἐζήτουν τὴν ὑπὸ τῶν ΚΕ ∠ γωνίαν τῇ ὑπὸ τῶν ∠ΕΘ γωνίᾳ ἴσην, καὶ ἔστιν ἴση ἡ ∠Κ τῇ ΚΕ, ὅτι ἄρα ἴση ἐστὶν ἡ ὑπὸ ΚΕ ∠ γωνία τῇ ὑπὸ Κ ∠Ε, ὅτι ἄρα καὶ ἡ ὑπὸ Κ ∠Ε τῇ ὑπὸ ∠ΕΘ ἴση ἐστίν , ὅτι ἄρα παράλληλός ἐστιν ἡ ∠Κ τῇ ΕΘ. Ἤχθω καὶ τῇ σΕ παράλληλος ἡ ΚΛ, καὶ ἐκβεβλήσθω ἡ Γ ∠ ἐπὶ τὸ Λ, καὶ ἐπεζεύχθω ἡ BΛ. ἐπεὶ οὖν ἡ μὲν ΚΛ τῇ ∠Ε ἐστὶν παράλληλος, ἡ δὲ ΚΗ τῇ EΖ, ζητεῖται δὲ καὶ ἡ Κ ∠ τῇ ΕΘ παράλληλος, ὅτι ἄρα διὰ τὸ ἰσογώνιον εἶναι τὸ μὲν ΚΛΗ τρίγωνον τῷ Κ ∠Ζ τριγώνῳ, τὸ δὲ ∠ΚΗ τῷ ΕΘΖ, ἔστιν ὡς μὲν ἡ ΛΗ πρὸς ΗK, ἡ ∠Ζ πρὸς ΖΕ, ὡς δὲ ἡ ΚΗ πρὸς Η ∠, ἡ ΕΖ πρὸς ΖΘ· ὅτι ἄρα καὶ ὡς ἡ ΛΗ πρὸς Η ∠, οὕτως ἡ ∠Ζ πρὸς ΖΘ (διʼ ἴσου γάρ)· ὅτι ἄρα καὶ ἁς ἡ Λ ∠ πρὸς τὴν ∠Η, οὕτως ἡ ∠Θ πρὸς τὴν ΘΖ (διελόντι γάρ). ὑπέκειτο δέ καὶ ὡς ἡ ∠Θ πρὸς ΘΖ, οὕτως ἡ ΑΗ πρὸς Η ∠ ὅτι ἄρα ἐστὶν ὡς ἡ Λ ∠ πρὸς ∠Η, οὕτως ἡ ∠Θ πρὸς ΘΖ , τουτέστιν ἡ ΑΗ πρὸς Η ∠· ὅτι ἄρα ἴση ἐστὶν ἡ Λ ∠ τῇ ΑΗ· ὃτι ἄρα καὶ ἡ ΛΑ τῇ ∠Η ἐστὶν ἴση. ἀλλὰ καὶ ἡ Ἀὲ τῇ Β ∠ ἐστὶν ἴση ὅτι ἄρα καὶ ἡ ΛΒ τῇ ΗΒ ἐστὶν ἴση. ἀλλὰ ἡ ΒΗ ἑκατέρᾳ τῶν Λ ∠ ΑΗ ἐστὶν ἴση ὅτι ἄρα καὶ ἡ ΒΛ τῇ Λ ∠ ἐστὶν ἴση. ἔστιν δέ· ἐπεὶ γάρ παράλληλός ἐστιν ἡ ΚΛ τῇ ∠Ε, καὶ ἔστιν ἴση ἡ ∠Κ τῇ ΚΕ, ἴση ἐστὶν καὶ ἡ ὑπὸ τῶν BΚΛ γωνία τῇ ὑπὸ τῶν ΛΚ ∠. ἐπεὶ οὖν ἴση ἐστὶν ἡ BΚ τῇ Κ ∠ καὶ γωνία ἡ ὑπὸ τῶν ΒΚΛ γωνίᾳ τῇ ὑπὸ τῶν ∠ΚΛ ἐστὶν ἴση, καὶ ἡ ΒΛ ἄρα τῇ Λ ∠ ἐστὶν ἴση.

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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/.