# Scholia in Euclidis Data Ad prop. LXX.

Scholia in Euclidem, Scholia in Euclidis Data, SCHOLIA.. Ancient Greek.

Canonical: https://interlinea.aeterna-institute.org/library/greek/scholia-in-euclidem/scholia-in-euclidis-data/Ad%20prop.%20LXX.

139. Ἀντιστρόφιον δύο πρὸ αὐτοῦ θεωρήμασιν. 140. Ἀντιστρόφιον τοῖς δύο ὁμοῦ τῷ τε ἑξηκόστῳ ὀγδόῳ καὶ τῷ ξθ΄ θεωρήματι. 141. P.132,4] ἐπʼεὐθείας ἄρα ἐστὶ καὶ ἡ ∠Β τῇ ΒΜ. ἐπεὶ γὰρ παράλληλός ἐστιν ἡ ΑΝ τῇ ∠Μ, αἱ ἐναλλὰξ γωνίαι αἱ ὑπὸ ∠ΒΓ, ΒΓΜ ἴσαι ἀλλήλαις εἰσίν. πάλιν ἐπεὶ παράλληλός ἐστιν ἡ ΜΒ τῇ ΑΓ, αἱ ὑπὸ ΜΒΓ, ΑΓΒ ἴσαι ἀλλήλαις εἰσίν. αἱ ἄρα ὑπὸ ΑΓΒ, ΒΓΝ ταῖς ὑπὸ ∠ΒΓ, ΓΒΜ ἴσαι εἰσίν. ὀρθαὶ δὲ αἱ ὑπὸ ΑΓΒ, BΓΝ· ὀρθαὶ ἄρα καὶ αἱ ὑπὸ ∠ΒΓ, ΓΒΜ. ἐὰν δὲ πρός τινι εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ καὶ τὰ ἑξῆς, ὡς ἐν τῷ α΄ τῶν στοιχείων (I, 14). 142. Ἔστι δὲ καὶ ἰσογώνιον p. 132, 6] ἐπεὶ γὰρ ἰσογώνιον κεῖται τὸ ΑΒ τῷ ΕΗ, ἴση ἐστὶν ἡ ὑπὸ ΑΓΒ τῇ πρὸς τῷ Ζ· ἀλλʼ ἡ ὑπὸ ΑΓΒ τῇ πρὸς τῷ Ν, ἡ ἐκτὸς τῇ ἐντός· καὶ ἡ πρὸς τῷ Ν ἄρα τῇ πρὸς τῷ Ζ ἴση. ὁμοίως καὶ αἱ λοιπαί. 143. Ἔστι δὲ καὶ ἡ ὑπὸ ΚΓΒ δοθεῖσα p. 132, 20] ἴση γάρ ἐστιν ἡ πρὸς τῷ Ζ δοθεῖσα. 144. Καθόλου γὰρ πάλιν, ἐὰν δύο τετραγώνων δύο γωνίαι ἴσαι ὦσιν, ἰσογώνια ἔσται τὰ παραλληλό- γραμμα. 145. Λόγος ἄρα ἐστὶ τοῦ ΓΛ πρὸς τὸ ΖΘ δοθείς p. 134, 6] μᾶλλον ἀληθῶς διὰ τοῦτο· ἐπεὶ γὰρ ἴση ἐστὶν ἡ ὑπὸ ΚΓΒ τῇ Ζ καὶ περὶ ἴσας γωνίας αἱ πλευραὶ λόγον ἔχουσι δεδομένον, διὰ τὸ νῦν πρῶτον δειχθὲν τοῦ ο΄ θεωρήματος λόγος ἐστὶ τοῦ ΓΛ πρὸς ΖΘ δοθείς.

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Scholia in Euclidem, Scholia in Euclidis Data, 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/tlg5022.tlg002.local001/.