ο΄.
ἴσας γωνίας τὰς πρὸς τοῖς Γ, Ζ ἢ περὶ ἀνίσους μέν, δεδομένας δέ, αἱ πλευραὶ πρὸς ἀλλήλας λόγον ἐχέτω- σαν δεδομένον, τουτέστι λόγος ἔστω τῆς μὲν ΑΓ πρὸς τὴν ΕΖ δοθείς, τῆς δὲ ΒΓ πρὸς τὴν ΖΗ· λέγω, ὅτι καὶ τοῦ ΓΔ πρὸς τὸ ΖΘ λόγος ἐστὶ δοθείς.
ἄρα τῇ ΒΜ ἐστιν ἐπʼ εὐθείας. καὶ ἴσον ἐστὶ τὸ ΒΝ τῷ ΖΘ· ἔστι δὲ καὶ ἰσογώνιον· τῶν ΒΝ, ΘΖ ἄρα ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας· ἔστιν ἄρα ὡς ἡ ΓΒ πρὸς τὴν ΖΗ, οὕτως ἡ ΖΕ πρὸς τὴν ΓΝ. λόγος δὲ τῆς ΓΒ πρὸς τὴν ΖΗ δοθείς·
λόγος ἄρα καὶ τῆς ΕΖ πρὸς τὴν ΓΝ δοθείς. τῆς δὲ ΕΖ πρὸς τὴν ΑΓ λόγος ἐστὶ δοθείς· καὶ τῆς ΑΓ ἄρα πρὸς τὴν ΓΝ λόγος ἐστὶ δοθείς· ὥστε καὶ τοῦ ΓΔ πρὸς τὸ ΓΜ λόγος ἐστὶ δοθείς. ἔστι δὲ τὸ ΓΜ τῷ ΖΘ ἔσον· λόγος ἄρα καὶ τοῦ ΓΔ πρὸς τὸ ΕΗ
δοθείς.
ἐπεὶ δοθεῖσά ἐστιν ἡ ὑπὸ τῶν ΑΓΒ, ἔστι δὲ καὶ ἡ ὑπὸ ΚΓΒ δοθεῖσα, καὶ λοιπὴ ἄρα ἡ ὑπὸ ΑΓΚ ἐστι δοθεῖσα. ἔστι δὲ καὶ ἡ ὑπὸ τῶν ΓΑΚ δοθεῖσα· καὶ λοιπὴ ἄρα ἡ ὑπὸ τῶν ΑΚΓ ἐστι δοθεῖσα· δέδοται ἄρα 2 3 τὸ ΑΓΚ τρίγωνον τῷ εἴδει· λόγος ἄρα ἐστὶ τῆς ΑΓ πρὸς τὴν ΓΚ δοθείς· τῆς δὲ ΑΓ πρὸς τὴν ΕΖ λόγος ἐστὶ δοθείς· καὶ τῆς ΓΚ ἄρα πρὸς τὴν ΕΖ λόγος ἐστὶ δοθείς. ἔστι δὲ καὶ τῆς ΓΒ πρὸς τὴν ΖΗ λόγος
δοθείς, καί ἐστιν ἴση ἡ ὑπὸ τῶν ΚΓΒ γωνία τῇ ὑπὸ τῶν ΕΖΗ· λόγος ἄρα ἐστὶ τοῦ ΓΛ πρὸς τὸ ΖΘ δοθείς. ἴσον δὲ τὸ ΓΛ τῷ ΓΔ· λόγος ἄρα ἐστὶ τοῦ ΓΔ πρὸς τὸ φῶ δοθείς.
Data. Euclid. ed. Johan Ludvig Heiberg, ed. Heinrich Menge. United States. 2016.
Sponsored by Harvard College Library.