Chapter 6

ιη΄. Τρίγωνον τὸ ΑΒΓ, καὶ τῇ ΒΓ παράλληλος ἤχθω ἡ Α∠, καὶ διήχθωσαν αἱ ∠Ε, ΖΗ, ἔστω δέ, ὡς τὸ ἀπὸ ΕΒ πρὸς τὸ ὑπὸ ΕΓΒ, οὕτως ἡ ΒΗ πρὸς τὴν ΗΓ· ὅτι, ἐὰν ἐπιζευχθῇ ἡ Β∠, γίνεται εὐθεῖα ἡ διὰ τῶν Θ, Κ, Γ. ἐπεί ἔστιν, ὡς τὸ ἀπὸ τῆς ΕΒ πρὸς τὸ ὑπὸ ΕΓΒ, οὕτως ἡ ΒH πρὸς ΗΓ, κοινὸς ἄρα προσκείσθω ὁ τῆς ΓΕ πρὸς ΕΒ λόγος ὁ αὐτὸς ὢν τῷ τοῦ ὑπὸ ΕΓΒ πρὸς τὸ ὑπὸ ΚΒΓ· δι᾿ ἴσου ἄρα ὁ τοῦ ἀπὸ ΕΒ πρὸς τὸ ὑπὸ ΕΒΓ λόγος, τουτέστιν ὁ τῆς ΕΒ πρὸς τὴν ΒΓ, ὁ αὐτός ἐστιν τῷ συνημμένῳ ἔκ τε τοῦ τῆς ΒΗ πρὸς ΗΓ καὶ τοῦ τοῦ ὑπὸ ΕΓΒ πρὸς τὸ ὑπὸ ΕΒΓ, ὅς ἔστιν ὁ αὐτὸς τῷ τῆς ΕΓ πρὸς ΕΒ· ὥστε ὁ τοῦ ἀπὸ ΕΒ πρὸς τὸ ὑπὸ ΕΒΓ συνῆπται ἔκ τε τοῦ ὃν ἔχει ἡ ΒΗ πρὸς ΗΓ καὶ τοῦ ὃν ἔχει ἡ ΕΓ πρὸς ΕΒ, ὅς ἐστιν ὁ αὐτὸς τῷ τοῦ ὑπὸ ΕΓ, ΒΗ πρὸς τὸ ὑπὸ ΕΒ, ΓΗ. ὡς δὲ ἡ ΕΒ πρὸς τὴν ΒΓ, οὕτως ἐστὶν διὰ τὸ προγεγραμμένον λῆμμα τὸ ὑπὸ ∠Ε, ΖΘ πρὸς τὸ ὑπὸ ∠Ζ, ΘΕ· καὶ ὡς ἄρα τὸ ὑπὸ ΓΕ, ΒΗ πρὸς τὸ ὑπὸ ΓΗ, ΕΒ, οὕτως ἐστὶν τὸ ὑπὸ ∠Ε, ΖΘ πρὸς τὸ ὑπὸ ∠Ζ, ΘΕ. εὐθεῖα ἄρα ἐστὶν ἡ διὰ τῶν Θ, Κ, Γ· τοῦτο γὰρ ἐν τοῖς πτωτικοῖς τῶν ἀναστροφίων.