ξδ΄.

Ἐὰν τρίγωνον ἀμβλεῖαν ἔχῃ γωνίαν δεδομένην, ᾦ μεῖζον δύναται ἡ τὴν ἀμβλεῖαν γωνίαν ὑποτείνουσα πλευρὰ τῶν τὴν ἀμβλεῖαν γωνίαν περιεχουσῶν πλευ- ρῶν, ἐκεῖνο τὸ χωρίον πρὸς τὸ τρίγωνον λόγον ἕξει δεδομένον. ἔστω τρίγωνον ἀμβλυγώνιον τὸ ΑΒΓ ἀμβλεῖαν γωνίαν ἔχον τὴν ὑπὸ τῶν ΑΒΓ δεδομένην, καὶ διήχθω ἐπʼ εὐθείας τῆς ΒΓ εὐθεῖα ἡ ΒΔ, καὶ ἤχθω ἀπὸ τοῦ Α ἐπὶ τὴν ΓΔ κάθετος ἡ ΑΔ· λέγω, ὅτι, ᾦ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΑΓ τῶν ἀπὸ τῶν ΑΒ, ΒΓ, τουτέστι τὸ δὶς ὑπὸ τῶν ΔΒ, ΒΓ ἐκεῖνο τὸ χωρίον πρὸς τὸ ΑΒΓ τρίγωνον λόγον ἕξει δεδομένον. ἐπεὶ γὰρ δοθεῖσά ἐστιν ἡ ὑπὸ ΑΒΓ, καὶ ἡ ὑπὸ τῶν ΑΒΔ δοθεῖσά ἐστιν. ἔστι δὲ καὶ ἡ ὑπὸ τῶν ΑΔΒ δοθεῖσα. καὶ λοιπὴ ἄρα ἡ ὑπὸ τῶν ΔΑΒ δο- θεῖσά ἐστιν. δέδοται ἄρα τὸ ΔΑΒ τρίγωνον τῷ εἴδει· λόγος ἄρα τῆς ΑΔ πρὸς τὴν ΔΒ δοθείς. καί ἐστιν ὡς ἡ ΑΔ πρὸς τὴν ΔΒ, οὕτως τὸ ὑπὸ τῶν ΑΔ, ΒΓ πρὸς τὸ ὑπὸ τῶν ΔΒ, ΒΓ· ὥστε καὶ τοῦ ὑπὸ τῶν ΔΑ, ΒΓ πρὸς τὸ ὑπὸ τῶν ΔΒ, ΒΓ λόγος ἐστὶ δοθείς· καὶ τοῦ δὶς ὑπὸ τῶν ΔΒ, ΒΓ ἄρα πρὸς τὸ ὑπὸ τῶν ΑΔ, ΒΓ λόγος ἐστὶ δοθείς. ἀλλὰ τοῦ ὑπὸ τῶν ΔΑ, ΒΓ πρὸς τὸ ΑΒΓ τρίγωνον λόγος ἐστὶ δοθείς· καὶ τοῦ δὶς ὑπὸ τῶν ΔΒΓ ἄρα πρὸς τὸ ΑΒΓ τρίγωνον λόγος ἐστὶ δοθείς. καί ἐστι τὸ δὶς ὑπὸ τῶν ΔΒ, ΒΓ, ᾧ μεῖζόν ἐστι τὸ ἀπὸ τῆς ΑΓ τῶν ἀπὸ τῶν ΑΒ, ΒΓ· ἐκεῖνο ἄρα τὸ χωρίον πρὸς τὸ ΑΒΓ τρίγωνον λόγον ἔχει δεδομένον.