πς΄.

Ἐὰν δύο εὐθεῖαι δοθὲν χωρίον περιέχωσιν ἐν δεδο. μένῃ γωνίᾳ, δύνηται δὲ ἡ ἑτέρα τῆς ἑτέρας δοθέντι μεῖζον ἢ ἐν λόγῳ, καὶ ἑκατέρα αὐτῶν ἔσται δοθεῖσα. δύο γὰρ εὐθεῖαι αἱ ΑΒ, ΒΓ δοθὲν χωρίον περι- εχέτωσαν τὸ ΑΓ ἐν δεδομένῃ γωνίᾳ τῇ ὑπὸ τῶν ΑΒΓ, τὸ δὲ ἀπὸ τῆς ΓΒ τοῦ ἀπὸ τῆς ΒΑ δοθέντι μεῖζον ἔστω ἢ ἐν λόγῳ· λέγω, ὅτι καὶ ἑκατέρα τῶν ΑΒ, ΒΓ ἐστι δοθεῖσα. ἐπεὶ γὰρ τὸ ἀπὸ τῆς ΓΒ τοῦ ἀπὸ τῆς ΒΑ δοθέντι μεῖζόν ἐστιν ἢ ἐν λόγῳ, ἀφῃρήσθω τὸ δοθὲν τὸ ὑπὸ τῶν ΓΒΔ· λοιποῦ ἄρα τοῦ ὑπὸ τῶν ΔΓΒ πρὸς τὸ ἀπὸ τῆς ΑΒ λόγος ἐστὶ δοθείς. καὶ ἐπεὶ δοθέν ἐστι τὸ ὑπὸ τῶν ΑΒΓ, ἔστι δὲ καὶ τὸ ὑπὸ τῶν ΓΒ, ΒΔ δοθέν, λόγος ἄρα ἐστὶ τοῦ ὑπὸ τῶν ΑΒ, ΒΓ πρὸς τὸ ὑπὸ τῶν ΓΒΔ δοθείς. ὡς δὲ τὸ ὑπὸ τῶν ΑΒΓ πρὸς τὸ ὑπὸ τῶν ΓΒ, ΒΔ, οὕτως ἡ ΑΒ πρὸς τὴν ΒΔ· ὥστε καὶ τῆς ΑΒ πρὸς τὴν ΒΔ λόγος ἐστὶ δοθείς· ὥστε καὶ τοῦ ἀπὸ τῆς ΑΒ πρὸς τὸ ἀπὸ τῆς ΒΔ λόγος ἐστὶ δοθείς. τοῦ δὲ ἀπὸ τῆς ΑΒ πρὸς τὸ ὑπὸ τῶν ΒΓΔ λόγος ἐστὶ δοθείς· καὶ τοῦ ὑπὸ τῶν ΒΓΔ ἄρα πρὸς τὸ ἀπὸ τῆς ΔΒ λόγος ἐστὶ δοθείς· ὥστε καὶ τοῦ τετράκις ὑπὸ τῶν ΒΓΔ πρὸς τὸ ἀπὸ τῆς ΒΔ λόγος ἐστὶ δοθείς· τοῦ τετράκις ὑπὸ τῶν ΒΓΔ ἄρα μετὰ τοῦ ἀπὸ τῆς ΒΔ πρὸς τὸ ἀπὸ τῆς ΒΔ λόγος ἐστὶ δοθείς. ἀλλὰ τὸ τετράκις ὑπὸ τῶν ΒΓΔ μετὰ τοῦ ἀπὸ τῆς ΒΔ τὸ ἀπὸ συναμφοτέρου ἐστὶ τῆς ΒΓ, ΓΔ. λόγος ἄρα ἐστὶ καὶ τοῦ ἀπὸ συναμφοτέρου τῆς ΒΓ, ΓΔ πρὸς τὸ ἀπὸ τῆς ΒΔ δοθείς· ὥστε καὶ συναμφοτέρου τῆς ΒΓΔ πρὸς τὴν ΒΔ λόγος ἐστὶ δοθείς· καὶ συν- θέντι ἄρα δύο τῶν ΓΒ πρὸς τὴν ΒΔ λόγος ἐστὶ δο- θείς· ὥστε καὶ μιᾶς τῆς ΓΒ πρὸς τὴν ΒΔ λόγος ἐστὶ δοθείς. ὡς δὲ ἡ ΓΒ πρὸς ΒΔ, οὕτως τὸ ὑπὸ τῶν ΓΒΔ πρὸς τὸ ἀπὸ τῆς ΒΔ· καὶ τοῦ ὑπὸ τῶν ΓΒΔ ἄρα πρὸς τὸ ἀπὸ τῆς ΒΔ λόγος ἐστὶ δοθείς. δοθὲν δὲ τὸ ὑπὸ τῶν ΓΒ, ΒΔ· δοθὲν ἄρα καὶ τὸ ἀπὸ τῆς ΒΔ· δοθεῖσα ἄρα ἐστὶν ἡ ΒΔ· ὥστε καὶ ἡ ΒΓ δο- θεῖσά ἐστιν· τῆς γὰρ ΓΒ πρὸς τὴν ΒΔ λόγος ἐστὶ δοθείς, καὶ δέδοται ἡ ΒΔ· καί ἐστι δοθὲν τὸ ΑΓ, καὶ δοθεῖσα ἡ Β γωνία· δοθεῖσα ἄρα ἐστὶ καὶ ἡ ΑΒ· ἑκατέρα ἄρα τῶν ΑΒ, ΒΓ δοθεῖσά ἐστιν.