# Scholia in Euclidis Data Ad prop. LXVII.

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.%20LXVII.

133. Ἐὰν ἰσοσκελοῦς τριγώνου ἀχθῇ τις εὐθεῖα, ὡς ἔτυχεν, ἐπὶ τὴν βάσιν, τὸ ἀπὸ τῆς καταχθείσης μετὰ τοῦ ὑπὸ τῶν τμημάτων τῆς βάσεως ἴσον ἐστὶ τῷ ἀπὸ μίας τῶν ἴσων πλευρῶν. ἔστω δὴ ἰσοσκελὲς τὸ ΑΒΓ ἴσην ἔχον τὴν ΑΒ τῇ ΑΓ, καὶ ἀπὸ τοῦ Α ἐπὶ τὴν ΒΓ ἤχθω τις εὐθεῖα, ὡς ἔτυχεν, ἡ Α∠. λέγω, ὅτι τὸ ἀπὸ τῆς Α∠ μετὰ τοῦ ὑπὸ τῶν Β∠, ∠Γ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΓ. ἡ Α∠ ἐπὶ τὴν ΒΓ ἤτοι κάθετός ἐστιν ἢ οὔ. ἔστω πρότερον κάθετος. καὶ ἐπεὶ εὐθεῖά τις ἡ ΒΓ τέτμηται δίχα κατὰ τὸ ∠, τὸ ἄρα ὑπὸ τῶν Γ∠, ∠Β ἴσον ἐστὶ τῷ ἀπὸ τῆς Β∠. κοινὸν προσκείσθω τὸ ἀπὸ τῆς Α∠· τὸ ἄρα ὑπὸ τῶν Γ∠, ∠Β μετὰ τοῦ ἀπὸ τῆς Α∠ ἴσον ἐστὶ τοῖς ἀπὸ τῶν Α∠, ∠Β. ἀλλὰ τοῖς ἀπὸ τῶν Α∠, ∠Β ἴσον ἐστὶ τὸ ἀπὸ ΑΒ τὸ ἄρα ὑπὸ τῶν Γ∠Β μετὰ τοῦ ἀπὸ τῆς Α∠ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΑΒ. ἀλλὰ δὴ μὴ ἔστω κάθετος ἡ Α∠, καὶ ἤχθω ἀπὸ τοῦ Α ἐπὶ τὴν ΒΓ κάθετος ἡ ΑΕ. καὶ ἐπεὶ εὐθεῖά τις τέτμηται εἰς μὲν ἴσα κατὰ τὸ Ε, εἰς δὲ ἄνισα κατὰ τὸ ∠, τὸ ἄρα ὑπὸ τῶν Γ∠Β μετὰ τοῦ ἀπὸ τῆς ∠Ε ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΕ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΑΕ· τὸ ἄρα ὑπὸ τῶν Γ∠Β μετὰ τῶν ἀπὸ τῶν ΑΕ∠ ἴσον ἐστὶ τοῖς ἀπὸ τῶν ΑΕΒ. καί ἐστιν ἴσον τοῖς ἀπὸ τῶν ΑΕ∠ τὸ ἀπὸ τῆς Α∠. τὸ ἄρα ὑπὸ τῶν Γ∠Β μετὰ τοῦ ἀπὸ Α∠ ἴσον ἐστὶ τοῖς ἀπδ τῶν ΑΕ, ΕΒ. καί ἐστι τοῖς ἀπὸ ΑΕ, ΕΒ τὸ ἀπὸ ΑΒ ἴσον. τὸ ἄρα ὑπὸ Γ∠Β μετὰ τοῦ ἀπὸ Α∠ ἴσον τῷ ἀπὸ ΑΒ. 134. Ἰσογώνια γάρ ἐστι τὰ ∠ΑΓ ∠ΒΕ τρίγωνα. 135. Καὶ ἐπεί ἐστιν ὡς ἡ ΒΑ p. 124, 18] παρ- άλληλος γάρ ἐστιν ἡ ΑΓ τῇ ΒΕ.

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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/.