20. Ad prop XCIII. Ἄλλως.
Διήχθω ἡ ΑΓ ἐπὶ τὸ Ε, καὶ κείσθω τῇ ΒΓ ἴση ἡ ΓΕ, καὶ ἐπεζεύχθωσαν αἱ ΕΒ, ΒΔ.
ἐπεὶ διπλῆ ἐστιν ἡ ὑπὸ τῶν ΑΓΒ ἑκατέρας τῶν ὑπὸ τῶν ΑΓΔ, ΓΒΕ, ἴση ἄρα ἐστὶν ἡ ὑπὸ τῶν ΓΒΕ γωνία τῇ ὑπὸ τῶν ΑΓΔ, τουτέστι τῇ ὑπὸ τῶν ΑΒΔ.
κοινὴ προσκείσθω ἡ ὑπὸ τῶν ΑΒΓ· ὅλη ἄρα ἡ ὑπὸ τῶν ΔΒΓ ὅλῃ τῇ ὑπὸ τῶν ΖΒE ἐστιν ἴση. ἔστι δὲ καὶ ἡ ὑπὸ τῶν ΓΑΒ τῇ ὑπὸ τῶν ΓΔΒ ἴση· λοιπὴ ἄρα ἡ ὑπὸ τῶν ΓΕΒ λοιπῇ τῇ ὑπὸ τῶν ΔΓΒ ἐστιν 1 2 ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΕΑΒ τρίγωνον τῷ ΓΔΒ τριγώνῳ· ἔστιν ἄρα, ὡς ἡ ΕΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΓΔ πρὸς τὴν ΔΒ· ἡ δὲ ΕΑ συναμφότερός ἐστιν ἡ ΑΓΒ· ὡς ἄρα συναμφότερος ἡ ΑΓΒ πρὸς τὴν ΑΒ,
οὕτως ἡ ΓΔ πρὸς τὴν ΒΔ· καὶ ἐναλλὰξ ἄρα, ὡς συν- αμφότερός ἐστιν ἡ ΑΓΒ πρὸς τὴν ΓΔ, οὕτως ἐστὶν ἡ ΑΒ πρὸς τὴν ΔΒ· λόγος δέ ἐστι τῆς ΑΒ πρὸς τὴν ΔΒ δοθείς· ἑκατέρα γὰρ αὐτῶν δοθεῖσα· λόγος ἄρα ἐστὶ καὶ συναμφοτέρου τῆς ΑΓΒ πρὸς τὴν ΓΔ
δοθείς.
κοινὴ προσκείσθω ἡ ὑπὸ τῶν ΑΒΓ· ὅλη ἄρα ἡ ὑπὸ τῶν ΔΒΓ ὅλῃ τῇ ὑπὸ τῶν ΖΒE ἐστιν ἴση. ἔστι δὲ καὶ ἡ ὑπὸ τῶν ΓΑΒ τῇ ὑπὸ τῶν ΓΔΒ ἴση· λοιπὴ ἄρα ἡ ὑπὸ τῶν ΓΕΒ λοιπῇ τῇ ὑπὸ τῶν ΔΓΒ ἐστιν 1 2 ἴση· ἰσογώνιον ἄρα ἐστὶ τὸ ΕΑΒ τρίγωνον τῷ ΓΔΒ τριγώνῳ· ἔστιν ἄρα, ὡς ἡ ΕΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΓΔ πρὸς τὴν ΔΒ· ἡ δὲ ΕΑ συναμφότερός ἐστιν ἡ ΑΓΒ· ὡς ἄρα συναμφότερος ἡ ΑΓΒ πρὸς τὴν ΑΒ,
οὕτως ἡ ΓΔ πρὸς τὴν ΒΔ· καὶ ἐναλλὰξ ἄρα, ὡς συν- αμφότερός ἐστιν ἡ ΑΓΒ πρὸς τὴν ΓΔ, οὕτως ἐστὶν ἡ ΑΒ πρὸς τὴν ΔΒ· λόγος δέ ἐστι τῆς ΑΒ πρὸς τὴν ΔΒ δοθείς· ἑκατέρα γὰρ αὐτῶν δοθεῖσα· λόγος ἄρα ἐστὶ καὶ συναμφοτέρου τῆς ΑΓΒ πρὸς τὴν ΓΔ
δοθείς.
καὶ ἐπεὶ ἰσογώνιόν ἐστι τὸ ΕΑΒ τρίγωνον τῷ ΖΒΔ τριγώνῳ, ἔστιν ἄρα, ὡς ἡ ΕΑ πρὸς τὴν ΑΒ, οὕτως ἡ ΒΔ πρὸς τὴν ΔΖ· ἡ δὲ ΕΑ συναμφότερός ἐστιν ἡ ΑΓΒ· ὡς ἄρα συναμφότερος ἡ ΑΓΒ πρὸς
τὴν ΑΒ, οὕτως ἡ ΒΔ πρὸς τὴν ΔΖ· τὸ ἄρα ὑπὸ συναμφοτέρου τῆς ΑΓΒ καὶ τῆς ΖΔ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΒ, ΒΔ· δοθὲν δέ ἐστι τὸ ὑπὸ τῶν ΑΒ, ΒΔ· δοθεῖσα γὰρ ἑκατέρα αὐτῶν· δοθὲν ἄρα ἐστὶ καὶ τὸ ὑπὸ συναμφοτέρου τῆς ΑΓΒ καὶ τῆς ΖΔ.
τὴν ΑΒ, οὕτως ἡ ΒΔ πρὸς τὴν ΔΖ· τὸ ἄρα ὑπὸ συναμφοτέρου τῆς ΑΓΒ καὶ τῆς ΖΔ ἴσον ἐστὶ τῷ ὑπὸ τῶν ΑΒ, ΒΔ· δοθὲν δέ ἐστι τὸ ὑπὸ τῶν ΑΒ, ΒΔ· δοθεῖσα γὰρ ἑκατέρα αὐτῶν· δοθὲν ἄρα ἐστὶ καὶ τὸ ὑπὸ συναμφοτέρου τῆς ΑΓΒ καὶ τῆς ΖΔ.
Data (demonstrationes alterae). Euclid. ed. Johan Ludvig Heiberg, ed. Heinrich Menge. United States. 2016.
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