ΠΑΠΠΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΣΥΝΑΓΩΓΗΣ Γ.

100 Καὶ ἐπεὶ παράλληλός ἐστιν ἡ ΚΛ τῇ ∠Γ, αἱ ὑπὸ ΛΚ∠ Κ∠Γ δυσὶν ὀρθαῖς ἴσαι εἰσίν. ὧν ὀρθὴ ἡ ὑπὸ Κ∠Γ· καὶ ἡ ὑπὸ ΛΚ∠ ἄρα ὀρθή ἐτιν· πρὸς ὀρθὰς ἄρα ἡ ΚΛ τῇ ΒΕ. καὶ ἐπεὶ ἐν ἡμικυκλίῳ τῷ ΒΛΕ πρὸς ὀρθὰς ἦκται ἡ ΛΚ μέση ἀνάλογόν ἐστιν ἡ ΛΚ τῶν ΕΚ ΚΒ· ἔστιν ἄρα ὡς τὸ ἀπὸ τῆς ΕΚ πρὸς τὸ ἀπὸ τῆς ΚΛ, οὕτως ἡ ΕΚ πρὸς ΚΒ. καὶ ἐπειδὴ παράλληλός ἐστιν ἡ μὲν ΒΖ τῇ ΛΕ, ἡ δὲ ΘΛ τῇ ΖΞ, ὅμοιόν ἐστι τὸ μὲν ΛΚΕ τρίγωνον τῷ ΒΚΘ, τὸ δὲ ΒΚΘ τῷ Β∠Ζ, καὶ ἔτι τὸ μὲν ΘΚΕ ὅμοιόν ἐστι τῷ Μ∠Ε, τὸ δὲ ΛΚΕ τῷ Ξ∠Ε· ἰσογώνιον ἄρα ἕκαστον ἔκάστῳ. ἔστιν ἄρα ὡς τὸ ἀπὸ τῆς ΕΚ πρὸς τὸ ἀπὸ τῆς ΚΛ, οὕτως τὸ ἀπὸ τῆς ΒΚ πρὸς τὸ ἀπὸ τῆς ΚΘ. ὡς δὲ τὸ ἀπὸ τῆς ΒΚ πρὸς τὸ ἀπὸ τῆς ΚΘ, οὕτως τὸ ἀπὸ τῆς Β∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ· ὡς ἄρα ἐστὶν τὸ ἀπὸ τῆς ΕΚ πρὸς τὸ ἀπὸ τῆς ΚΛ, οὕτως ἐστὶν τὸ ἀπὸ τῆς Β∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ. ἀλλʼ ὡς τὸ ἀπὸ τῆς ΕΚ πρὸς τὸ ἀπὸ τῆς ΚΛ, οὕτως ἡ ΕΚ πρὸς τὴν ΚΒ· καὶ ὡς ἄρα ἡ ΕΚ πρὸς τὴν ΚΒ, οὕτως τὸ ἀπὸ τῆς Β∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ. πάλιν ἐπεὶ ὅμοιόν ἐστι τὸ μὲν ΛΚΕ τρίγωνον τῷ ΒΚΘ, τὸ δὲ ΘΚΕ τῷ Μ∠Ε, καὶ τὸ ΛΚΕ τῷ Ξ∠Ε τριγόνῳ διὰ τὸ τὰς παραλλήλους ἰσογονία αὐτὰ ποιεῖν, ἔστιν ὡς μὲν ἡ ΕΚ πρὸς ΚΛ, οὕτως ἡ ΒΚ πρὸς ΚΘ, καὶ ἐναλλὰξ ὡς ἡ ΕΚ πρὸς τὴν ΚΒ, οὕτως ἡ ΛΚ πρὸς τὴν ΚΘ, ὡς δὲ ἡ ΕΚ πρὸς ΚΛ, οὕτως ἡ Ε∠ πρὸς τὴν ∠Ξ, καὶ ἐναλλὰξ ὡς ἡ Ε πρὸς Ε∠, οὕτως ἡ ΚΑ πρὸς ∠Ξ. ὁμοίως καί, ἐπεὶ ὡς ἡ ΕΚ πρὸς τὴν ΚΘ, οὕτως ἡ Ε∠ πρὸς ∠Μ, καὶ ἐναλλὰξ ὡς ἡ ΚΕ πρὸς Ε∠, οὕτως ἡ ΚΘ πρὸς τὴν ∠Μ, διʼ ἴσου ἄρα ἐστὶν ὡς ἡ ΚΛ πρὸς τὴν ∠Ξ, οὕτως ἡ ΘΚ πρὸς τὴν Μ∠, καὶ ἐναλλὰξ ὡς ἡ ΛΚ πρὸς τὴν ΚΘ, οὕτως ἡ Ξ∠ πρὸς ∠Μ. ἀλλʼ ὡς ἡ ΛΚ πρὸς ΚΘ, οὕτως ἡ ΕΚ πρὸς ΚΒ· καὶ ὡς ἄρα ἡ ΕΚ πρὸς τὴν ΚΒ, οὕτως ἡ Ξ∠ πρὸς τὴν ∠Μ. ἴση δὲ ἡ Ξ∠ τῇ ∠Ζ· ἔστιν ἄρα ὡς ἡ ΕΚ πρὸς τὴν ΚΒ, οὕτως ἡ Ζ∠ πρὸς τὴν ∠Μ. ἀλλʼ ὡς ἡ ΕΚ πρὸς τὴν ΚΒ, οὕτως τὸ ἀπὸ τῆς Β∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ· καὶ ὡς ἄρα τὸ ἀπὸ τῆς Β∠ πρὸς τὸ ἀπὸ τῆς ∠Ζ, οὕτως ἡ Ζ∠ πρὸς τὴν ∠Μ.