ΠΑΠΠΟΥ ΑΛΕΞΑΝΔΡΕΩΣ ΣΥΝΑΓΩΓΗΣ Ε.

86 νγ΄ (ι΄). Πάλιν ἔστω ἡμικύκλιον τὸ ΑΒΓ οὗ κέντρον τύ Ζ, καὶ πενταπλάσιον τὸ ἀπὸ ΓΖ τοῦ ἀπὸ ΕΖ, καὶ τῇ ΑΓ πρὸς ὀρθὰς ἡ ΒΕ, καὶ ἐπιζευχθεῖσα ἡ ΒΓ ἄκρον καὶ μέσον λόγον τετμήσθω τῷ Δ, καὶ ἔστω μείζων ἡ B∠· ὅτι τὰ ἀπὸ ΓB Β∠ πενταπλάσια τοῦ ἀπὸ ΓΕ. Κείσθω τῇ ΕΖ ἴση ἡ ΖΓ· ἡ ΗΓ ἄρα ἄκρον καὶ μέσον λόγον τέτμηται τῷ Ε, καὶ τὸ μεῖζον τμῆμά ἐστιν ἡ ΕΗ. καὶ ἐπεὶ τὸ ὑπὸ ΗΓΕ ἴσον ἐστὶν τῷ ἀπὸ ΕΗ, ἴση δέ ἐστιν ἡ ΕΓ τῇ ΑΗ (ὅτι καὶ ἡ ΕΖ τῇ ΖΗ), ἴσον ἄρα τὸ ὑπὸ ΑΕΓ τῷ ἀπὸ ΕΗ. καὶ ἔστιν ὡς μὲν τὸ ἀπὸ ΕΗ, τουτέστιν τὸ ὑπὸ ΑΕΓ, πρὸς τὸ ἀπὸ ΕΓ, οὕτως τὸ ἀπὸ ΒΓ πρὸς τὸ ἀπὸ Β∠ (ἐπεὶ καὶ ὡς ἡ ΗΕ πρὸς ΕΓ, ἡ ΓΒ πρὸς Β∠). ὡς δὲ τὸ ὑπὸ ΑΕΓ πρὸς τὸ ἀπὸ ΕΓ, οὕτως ΕΑ πρὸς ΕΓ καὶ ὡς ἄρα ἡ ΑΕ πρὸς ΕΓ, τὸ ἀπὸ ΓΒ πρὸς τὸ ἀπὸ Β∠. πλάσια τοῦ ἀπὸ ΒΘ, τὰ ἄρα ἀπὸ ΒΚ ΚΜ τριπλάσια τοῦ ἀπὸ ΖΗ.