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τὸ ὑπὸ μέσων μήκει συμμέτρων εὐθειῶν κατά τινα τῶν εἰρημένων τρόπων περιεχόμενον ὀρθογώνιον μέσον ἐστίν. ὑπὸ γὰρ μέσων μήκει συμμέτρων εὐθειῶν τῶν ΑΒ, ΒΓ περιεχέσθω ὀρθογώνιον τὸ ΑΓ: λέγω, ὅτι τὸ
5ΑΓ μέσον ἐστίν. Ἀναγεγράφθω γὰρ ἀπὸ τῆς ΑΒ τετράγωνον τὸ ΑΔ: μέσον ἄρα ἐστὶ τὸ ΑΔ. καὶ ἐπεὶ σύμμετρός ἐστιν ἡ ΑΒ τῇ ΒΓ μήκει, ἴση δὲ ἡ ΑΒ τῇ ΒΔ, σύμμετρος ἄρα ἐστὶ καὶ ἡ ΔΒ τῇ ΒΓ
10μήκει: ὥστε καὶ τὸ ΔΑ τῷ ΑΓ σύμμετρόν ἐστιν. μέσον δὲ τὸ ΔΑ: μέσον ἄρα καὶ τὸ ΑΓ: ὅπερ ἔδει δεῖξαι.
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