# Scholia in Euclidis Data Ad prop. LXVII demonstr. alt.

Scholia in Euclidem, Scholia in Euclidis Data, APPENDIX SCHΟLIΟRUM.. Ancient Greek.

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

16. Ὥστε καὶ τοῦ ὑπὸ τῶν ΕΓ∠ p. 206, 13] σχόλιον. ἐκ τῶν λαμβανομένων τῇ Γ∠ τῇ αὐτῇ ἀποδείξει τῇ ἐπὶ τοῦ ξδ΄ χρησόμεθα. ἐκθέμενοι εὐθέως τὴν αβ καὶ τῇ μὲν ΕΓ ἴσην τὴν αγ, τῇ δὲ ΑΖ τὴν γβ καὶ πρὸς ὀρ- θὰς ἀπὸ τοῦ γ τὴν γδ ἴσην οὖσαν τῇ Γ∠ καὶ τὰ ἑξῆς ὡς ἐν τῷ ξδ΄ θεωρήματι. 17. Σχόλιον. ὡς γὰρ ἡ ΕΓ πρὸς ΑΖ, οὕτως τὸ ὑπὸ τῶν ΕΓ∠ πρὸς τὸ ὑπὸ τῶν ΑΖ, Γ∠. 18. Τοῦ δὲ ὑπὸ τῶν ΑΖ, Γ∠ πρὸς τὸ ΑΓ∠ τρίγωνον p. 206, 15] διπλάσιον γάρ, φησίν, ἔστιν αὐτοῦ. πῶς; ἐκκείσθω τις εὐθεῖα ἡ ηθ, καὶ κείσθω τῇ μὲν Γ∠ ἴση ἡ ηθ, τῇ δὲ ΑΖ πρὸς ὀρθὰς ἀχθεῖσα ἡ ηκ, καὶ συμπεπληρώσθω τὸ κθ παρ- αλληλόγραμμον, καὶ ἔστω διαγώνιος ἡ θκ ἀντὶ τῆς Α∠· τὸ ἄρα ὑπὸ τῶν θη, ηκ ἐστι τὸ κθ, καί ἐστι δι᾿ αὐτοῦ ἡ θκ· διπλάσιον ἄρα ἐστὶ τοῦ κηθ τριγώνου· ἐπί τε γὰρ τῆς αὐτῆς βάσεώς ἐστι τῆς ηθ καὶ ἐν ταῖς αὐταῖς παρ- αλλήλοις ταῖς ηθ, κλ· καί ἐστι τὸ μὲν θκ παραλληλό- γραμμον ἴσον τῷ ὑπὸ τῶν ΑΖ, Γ∠, τὸ δὲ κηθ ἴσον τῷ ΑΓ∠ τριγώνῳ· διπλάσιον ἄρα τὸ ὑπὸ τῶν ΑΖ, Γ∠ τοῦ ΑΓ∠ τριγώνου. 19. Πῶς τὸ ὑπὸ τῶν ΑΖ, Γ∠ τοῦ ΑΓ∠ τριγώνου διπλάσιόν ἐστιν; δείξομεν οὕτως. ἤχθω διὰ τοῦ Α τῇ Γ∠ παράλληλος ἡ ΑΗ καὶ διὰ τοῦ Η τῇ ΑΖ παράλληλος ἡ ΗΘ. δύο ἄρα παραλληλόγραμμά ἐστι τὰ ΑΘ, Α∠ (ὑπό- κειται γὰρ καὶ ἡ ΑΓ τῇ Β∠ παράλληλος) ἐπὶ τῆς ΑΗ καὶ ἐν ταῖς αὐταῖς παραλλήλοις ταῖς ΑΗ, Ζ∠· ἴσον ἄρα τὸ ΑΘ παραλληλό- γραμμον τῷ Α∠ παραλληλογράμμῳ. καὶ ἐπεὶ τὸ ὑπὸ τῶν ΑΖ, ΑΗ ἐστι τὸ ΑΘ, ἴση δὲ ἡ ΑΗ τῇ Γ∠, καὶ τὸ ἄρα ὑπὸ τῶν ΑΖ, Γ∠ ἐστι τὸ ΑΘ διπλάσιον δὲ τὸ ΑΘ τοῦ ΑΓ∠ τριγώνου, ἐπεὶ καὶ τὸ Α∠· τὸ ἄρα ὑπὸ τῶν ΑΖ, Γ∠ διπλάσιόν ἐστι τοῦ ΑΓ∠ τριγώνου.

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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/.