# Synagoge 3.16

Pappus Alexandrinus, Synagoge, Book 3. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/pappus-alexandrinus/synagoge/3.16

16 ιβ΄ Ἕστω ἡμικύκλιον τὸ ΑΒΓ, καὶ κεκλάσθω ἡ ΓΒΑ, καὶ διήχθω ἡ Γ∠, καὶ ἴση ἔστω ἡ ΓΒ συναμφοτέρῳ τῇ AB △Γ, καὶ κάθετοι ἤχθωσαν ἐπὶ τὴv AΓ αἱ ΒΕ △Ζ ὅτι ἡ AΖ διπλασίων ἐστὶν τῆς BΕ. Κείσθω γὰρ τῇ μὲν AΕ ἴση ἡ ΕΗ, τῇ δὲ ΑΒ ἴση ἡ ΒΘ , καὶ ἐπεζεύχθωσαν εὐθεῖαι αἱ ΑΘ ΘΗ ΘΖ, καὶ κάθετος ἤχθω ἡ ΘΚ, καὶ ἐπεζεύχθω ἡ ΒΚ. ἐπεὶ ἡ ΓΒ ἴση ἐστὶν συναμφοτέρᾳ τῇ ΑΒ ∠Γ, ὧν ἡ ΒΘ τῇ BΑ ἐστὶν ἴση, λοιπὴ ἄρα ἡ ΘΓ λοιπῇ τῇ Γ ∠ ἐστὶν ἴση καὶ τὸ ἀπὸ τῆς Γ ∠ ἄρα ἴσον ἐστὶν τῷ ἀπὸ τῆς ΓΘ. τῷ δὲ ἀπὸ τῆς ∠Γ ἴσον ἐστὶν τὸ ὑπὸ τῶν ΑΓΖ· καὶ τὸ ὑπὸ τῶν ΑΓΖ ἄρα ἴσον ἐστὶν τῷ ἀπὸ τῆς ΓΘ· ἴση ἄρα ἐστὶν ἡ ὑπὸ τῶν ΖΘΓ γωνία τῇ ὑπὸ τῶν ΘΑΗ γωνίᾳ. πάλιν ἐπεὶ τὸ ὑπὸ τῶν Γ ΑΕ ἴσον ἐστὶν τῷ ἀπὸ τῆς ΑΒ, καὶ τὸ δὶς ἄρα ὑπὸ τῶν Γ ΑΕ, τουτέστιν τὸ ὑπὸ τῶν Γ ΑΗ, ἴσον ἐστὶν τῷ δὶς ἀπὸ τῆς ΑΒ, τουτέστιν τῷ ἀπὸ τῆς ΑΘ ἴση ἄρα ἐστὶν ἡ ὑπὸ τῶν ΑΘΗ γωνία τῇ ὑπὸ τῶν ΘΓΖ γωνίᾳ. ἔστιν δὲ καὶ ἡ ὑπὸ τῶν ΘΑΗ ἴση τῇ ὑπὸ τῶν ΖΘΓ· λοιπὴ ἄρα ἡ ὑπὸ τῶν ΑΗΘ λοιπῇ τῇ ὑπὸ τῶν ΘΖΓ ἐστὶν ἴση καὶ ἡ ὑπὸ ΘΗΖ ἄρα τῇ ὑπὸ ΘΖΗ ἐστὶν ἴση. καὶ κάθετος ἦκται ἡ ΘΚ· ἴση ἄρα ἐστὶν ἡ ΖΚ τῇ ΚΗ. καὶ ἐπεὶ ὀρθή ἐστιν ἑκατέρα τῶν ὑπὸ τῶν ΑΒΘ ΑΚΘ, ἐν κύκλῳ ἐστὶν τὸ ΑΒΘΚ τετράπλευρον ἴση ἄρα ἐστὶν ἡ ὑπὸ τῶν ΒΘΑ γωνία τῇ ὑπὸ τῶν ΒΚΑ. ἡμίσους δέ ἐστιν ἡ ὑπὸ τῶν ΒΘΑ· ἡμίσους ἅρα ἐστὶν καὶ ἡ ὑπὸ τῶν ΒΚΑ. ὀρθὴ δέ ἐστιν ἡ ὑπὸ τῶν ΒΕΚ· ἴση ἄρα ἐστὶν ἡ ΒΕ τῇ ΕΚ. τῆς δὲ ΕΚ διπλῆ ἐστιν ἡ ΑΖ (ἐπείπερ ἡ μὲν ΑΕ τῇ ΕΗ ἐστὶν ἴση, ἡ δὲ ΖΚ τῇ ΚΗ)· καὶ τῆς ΕΒ ἄρα διπλῆ ἐστιν ἡ ΑΖ, ὅπερ: ~

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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/.