Chapter 6

κθ΄. Τμήματος δοθέντος τοῦ ἐπὶ τῆς ΑΒ κλάσαι εὐθεῖαν θεῖαν τὴν ΑΓΒ ἐν λόγῳ τῷ δοθέντι. γεγονέτω, καὶ διήχθω ἀπὸ τοῦ Γ ἐφαπτομένη ἡ Γ∠ ὡς ἄρα τὸ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΒΓ, οὕτως ἡ Α∠ πρὸς ∠Β. λόγος δὲ τοῦ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΓΒ, δοθείς· ὥστε καὶ ὁ τῆς Α∠ πρὸς τὴν Β∠ δοθείς. καί ἐστιν δύο δοθέντα τὰ Α, Β· δοθὲν ἄρα ἐστὶν τὸ ∠· ὥστε καὶ τὸ Γ δοθέν. Συντεθήσεται δὴ τὸ πρόβλημα οὕτως. ἔστω τὸ μὲν τμῆμα τὸ ΑΒΓ, ὁ δὲ λόγος ὁ τῆς πρὸς τὴν Ζ, καὶ πεποιήσθω, ὡς τὸ ἀπὸ Ε πρὸς τὸ ἀπὸ Ζ, οὕτως ἡ Α∠ πρὸς τὴν ∠Β, καὶ ἤχθω ἐφαπτομένη ἡ ∠Γ, καὶ ἐπεζεύχθωσαν αἱ ΑΓ, ΓΒ. λέγω, ὅτι αἱ ΑΓ, ΓΒ ποιοῦσι τὸ πρόβλημα. ἐπεὶ γάρ ἐστιν, ὡς τὸ ἀπὸ Ε πρὸς τὸ ἀπὸ Ζ, οὕτως ἡ Α∠ πρὸς τὴν ∠Β, ὡς δὲ ἡ Α∠ πρὸς τὴν ∠Β, οὕτως τὸ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΓΒ διὰ τὸ ἔφάπτεσθαι τὴν ΓΔ, καὶ ὡς ἄρα τὸ ἀπὸ Ε πρὸς τὸ ἀπὸ Ζ, οὕτως τὸ ἀπὸ ΑΓ πρὸς τὸ ἀπὸ ΓΒ ὥστε καί, ὡς ἡ πρὸς τὴν Ζ, οὕτως ἡ ΑΓ πρὸς τὴν ΓΒ. ἡ ΑΓΒ ἄρα ποιεῖ τὸ πρόβλημα.