Ancient Greek · ~15 min
14. Αd prop LXVII. Ἄλλως.
Euclid, Data (demonstrationes alterae) 14. Αd prop LXVII. Ἄλλως..1
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Ἤτοι γὰρ ἡ Α γωνία ὀρθή ἐστιν ἢ ὀξεῖα ἢ ἀμβλεῖα. ἔστω πρότερον ὀρθή· τὸ ἄρα ἀπὸ συναμφοτέρου τῆς ΒΑΓ τοῦ ἀπὸ τῆς ΒΓ ὑπερέχει τῷ δὶς ὑπὸ τῶν ΒΑΓ. καί ἐστι τοῦ δὶς ὑπὸ τῶν ΒΑΓ πρὸς τὸ ΑΒΓ τρίγωνον λόγος δοθείς. ἔστω δὴ ὀξεῖα ἡ ὑπὸ τῶν ΒΑΓ, καὶ ἤχθω ἀπὸ τοῦ Γ ἐπὶ τὴν ΑΒ κάθετος ἡ ΓΔ. ἐπεὶ ὀξυγώνιόν ἐστι τὸ ΑΒΓ τρίγωνον, καὶ κάθετος ἦκται ἡ ΓΔ, τὰ ἄρα ἀπὸ τῶν ΒΑΓ ἴσα ἐστὶ τῷ τε ἀπὸ τῆς ΒΓ καὶ τῷ δὶς ὑπὸ τῶν ΒΑΔ. κοινὸν προσκείσθω τὸ δὶς ὑπὸ τῶν ΒΑΓ· τὰ ἄρα ἀπὸ τῶν ΒΑΓ μετὰ τοῦ δὶς ὑπὸ τῶν ΒΑΓ, ὅπερ ἐστὶ τὸ ἀπὸ συναμφοτέρου τῆς ΒΑΓ, ἴσα ἐστὶ τῷ τε ἀπὸ τῆς ΒΓ καὶ τῷ δὶς ὑπὸ τῶν ΒΑΔ καὶ ἔτι τῷ δὶς ὑπὸ τῶν ΒΑΓ, τουτέστι τῷ δὶς ὑπὸ συναμφοτέρου τῆς ΓΑΔ καὶ τῆς ΑΒ· ὥστε τὸ ἀπὸ συναμφοτέρου τῆς ΒΑΓ μεῖζόν ἐστι τοῦ ἀπὸ τῆς ΒΓ τῷ δὶς ὑπὸ συναμφοτέρου τῆς ΔΑΓ καὶ τῆς ΒΑ. καὶ ἐπεὶ δοθεῖσά ἐστιν ἡ ὑπὸ τῶν ΒΑΓ γωνία, ἔστι δὲ καὶ ἡ ὑπὸ τῶν ΑΔΓ γωνία δοθεῖσα, καὶ λοιπὴ ἄρα ἡ ὑπὸ τῶν ΔΓΑ ἐστι δοθεῖσα· δέδοται ἄρα τὸ ΑΔΓ τρίγωνον τῷ εἴδει· λόγος ἄρα ἐστὶ τῆς ΑΔ πρὸς τὴν ΑΓ δοθείς· ὥστε καὶ συναμφοτέρου τῆς ΔΑΓ πρὸς τὴν ΑΓ λόγος ἐστὶ δοθείς· καὶ τοῦ ὑπὸ συναμφοτέρου ἄρα τῆς ΔΑΓ καὶ τῆς Α Β πρὸς τὸ ὑπὸ τῶν ΒΑΓ λόγος ἐστὶ δοθείς, καὶ τοῦ δὶς ὑπὸ συναμφοτέρου τῆς ΔΑΓ καὶ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΒΑΓ λόγος ἐστὶ δοθείς. τοῦ δὲ ὑπὸ τῶν ΒΑΓ πρὸς τὸ ΒΑΓ τρίγωνον λόγος ἐστὶ δοθείς διὰ τὸ δοθεῖσαν εἶναι τὴν ὑπὸ τῶν ΒΑΓ γωνίαν· καὶ τοῦ δὶς ὑπὸ συναμφοτέρου τῆς ΔΑΓ καὶ τῆς ΑΒ ἄρα πρὸς τὸ ΑΒΓ τρίγωνον λόγος ἐστὶ δοθείς. ἀλλὰ δὴ ἔστω ἀμβλεῖα ἡ ὑπὸ ΒΑΓ, καὶ ἐκβληθείσης τῆς ΒΑ ἤχθω ἐπʼ αὐτὴν κάθετος ἡ ΓΕ, καὶ κείσθω τῇ ΑΕ ἴση ἡ ΑΖ. ἐπεὶ οὖν ἀμβλεῖά ἐστιν ἡ ὑπὸ ΒΑΓ γωνία, καὶ κάθετος ἦκται ἡ ΓΕ, τὰ ἄρα ἀπὸ τῶν ΒΑ, ΑΓ μετὰ τοῦ δὶς ὑπὸ τῶν ΒΑΕ, τουτέστι τοῦ δὶς ὑπὸ τῶν ΒΑΖ, ἴσα ἐστὶ τῷ ἀπὸ τῆς ΒΓ. κοινὸν προσκείσθω τὸ δὶς ὑπὸ τῶν, ΒΑΓ· τὰ ἄρα ἀπὸ τῶν ΒΑΓ μετὰ τοῦ δὶς ὑπὸ τῶν ΒΑΓ, τουτέστι τὸ ἀπὸ συναμφοτέρου τῆς ΒΑΓ μετὰ τοῦ δὶς ὑπὸ τῶν ΒΑΖ ἴσα ἐστὶ τῷ ἀπὸ, τῆς ΒΓ μετὰ τοῦ δὶς ὑπὸ τῶν ΒΑΓ· κοινὸν ἀφῃρήσθω τὸ δὶς ὑπὸ τῶν ΒΑΖ· τὸ ἄρα ἀπὸ συν- αμφοτέρου τῆς ΒΑΓ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΓ καὶ τῷ δὶς ὑπὸ τῶν ΒΑ, ΓΖ· ὥστε τὸ ἀπὸ συναμφο- τέρου τῆς ΒΑΓ τοῦ ἀπὸ τῆς ΒΓ ὑπερέχει τῷ δὶς ὑπὸ τῶν ΒΑ, ΓΖ. καὶ ἐπεὶ δοθεῖσά ἐστιν ἡ ὑπὸ ΒΑΓ γωνία, καὶ ἡ ὑπὸ ΕΑΓ ἄρα δοθεῖσά ἐστιν. ἀλλὰ καὶ ἡ ὑπὸ ΓΕΑ δοθεῖσα· καὶ λοιπὴ ἄρα ἡ ὑπὸ ΑΓΕ δοθεῖσά ἐστιν· δέδοται ἄρα τὸ ΑΚΓ τρίγωνον τῷ εἴδει. λόγος ἄρα τῆς ΓΑ πρὸς τὴν ΑΕ δοθείς, τουτέστι πρὸς τὴν ΑΖ· ὥστε καὶ τῆς ΑΓ πρὸς τὴν ΓΖ λόγος ἐστὶ δοθείς. τῆς δὲ ΑΓ πρὸς τὴν ΓΕ λόγος ἐστὶ δοθείς· καὶ τῆς ΕΓ ἄρα πρὸς τὴν ΓΖ λόγος ἐστὶ δοθείς· ὥστε καὶ τοῦ ὑπὸ τῶν ΕΓ, ΑΒ πρὸς τὸ ὑπὸ τῶν ΓΖ, ΑΒ λόγος ἐστὶ δοθείς. τοῦ δὲ ὑπὸ τῶν ΑΒ, ΓΕ πρὸς τὸ ΑΒΓ τρίγωνον λόγος ἐστὶ δοθείς· ὥστε καὶ τοῦ δὶς ὑπὸ ΓΖ, ΒΑ πρὸς τὸ ΑΒΓ τρί- γωνον λόγος ἐστὶ δοθείς. καί ἐστι τὸ δὶς ὑπὸ τῶν ΖΓ, ΒA, ᾧ μεῖζόν ἐστι τὸ ἀπὸ συναμφοτέρου τῆς ΒΑΓ τοῦ ἀπὸ τῆς ΒΓ· ᾧ ἄρα μεῖζόν ἐστι τὸ ἀπὸ συναμφο- τέρου τῆς ΒΑΓ τοῦ ἀπὸ τῆς ΒΓ, ἐκεῖνο τὸ χωρίον πρὸς τὸ τρίγωνον λόγον ἔχει δεδομένον.