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τὸ ἀπὸ μέσης παρὰ ῥητὴν παραβαλλόμενον πλάτος ποιεῖ ῥητὴν καὶ ἀσύμμετρον τῇ, παρ᾽ ἣν παράκειται, μήκει.

ἔστω μέση μὲν Α, ῥητὴ δὲ ΓΒ, καὶ τῷ ἀπὸ τῆς Α
5ἴσον παρὰ τὴν ΒΓ παραβεβλήσθω χωρίον ὀρθογώνιον τὸ ΒΔ πλάτος ποιοῦν τὴν ΓΔ: λέγω, ὅτι ῥητή ἐστιν ΓΔ καὶ ἀσύμμετρος τῇ ΓΒ μήκει.

ἐπεὶ γὰρ μέση ἐστὶν Α, δύναται χωρίον περιεχόμενον ὑπὸ ῥητῶν δυνάμει μόνον συμμέτρων. δυνάσθω τὸ ΗΖ.
10δύναται δὲ καὶ τὸ ΒΔ: ἴσον ἄρα ἐστὶ τὸ ΒΔ τῷ ΗΖ. ἔστι δὲ αὐτῷ καὶ ἰσογώνιον: τῶν δὲ ἴσων τε καὶ ἰσογωνίων παραλληλογράμμων ἀντιπεπόνθασιν αἱ πλευραὶ αἱ περὶ τὰς ἴσας γωνίας: ἀνάλογον ἄρα ἐστὶν ὡς ΒΓ πρὸς τὴν ΕΗ, οὕτως ΕΖ πρὸς τὴν ΓΔ. ἔστιν ἄρα καὶ ὡς τὸ ἀπὸ τῆς
10ΒΓ πρὸς τὸ ἀπὸ τῆς ΕΗ, οὕτως τὸ ἀπὸ τῆς ΕΖ πρὸς τὸ ἀπὸ τῆς ΓΔ. σύμμετρον δέ ἐστι τὸ ἀπὸ τῆς ΓΒ τῷ ἀπὸ τῆς ΕΗ: ῥητὴ γάρ ἐστιν ἑκατέρα αὐτῶν: σύμμετρον ἄρα ἐστὶ καὶ τὸ
20ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς ΓΔ. ῥητὸν δέ ἐστι τὸ ἀπὸ τῆς ΕΖ: ῥητὸν ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς ΓΔ: ῥητὴ ἄρα ἐστὶν ΓΔ. καὶ ἐπεὶ ἀσύμμετρός ἐστιν ΕΖ τῇ ΕΗ μήκει: δυνάμει γὰρ μόνον εἰσὶ σύμμετροι: ὡς δὲ ΕΖ πρὸς τὴν ΕΗ, οὕτως τὸ ἀπὸ τῆς ΕΖ
25πρὸς τὸ ὑπὸ τῶν ΖΕ, ΕΗ, ἀσύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς ΕΖ τῷ ὑπὸ τῶν ΖΕ, ΕΗ. ἀλλὰ τῷ μὲν ἀπὸ τῆς ΕΖ σύμμετρόν ἐστι τὸ ἀπὸ τῆς ΓΔ: ῥηταὶ γάρ εἰσι δυνάμει: τῷ δὲ ὑπὸ τῶν ΖΕ, ΕΗ σύμμετρόν ἐστι τὸ ὑπὸ τῶν ΑΓ, ΓΒ: ἴσα γάρ ἐστι τῷ ἀπὸ τῆς Α: ἀσύμμετρον ἄρα ἐστὶ καὶ τὸ ἀπὸ
30τῆς ΓΔ τῷ ὑπὸ τῶν ΔΓ, ΓΒ. ὡς δὲ τὸ ἀπὸ τῆς ΓΔ πρὸς τὸ ὑπὸ τῶν ΔΓ, ΓΒ, οὕτως ἐστὶν ΔΓ πρὸς τὴν ΓΒ: ἀσύμμετρος ἄρα ἐστὶν ΔΓ τῇ ΓΒ μήκει. ῥητὴ ἄρα ἐστὶν ΓΔ καὶ ἀσύμμετρος τῇ ΓΒ μήκει: ὅπερ ἔδει δεῖξαι.

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