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ἐὰν χωρίον περιέχηται ὑπὸ ἀποτομῆς καὶ τῆς ἐκ δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά τέ ἐστι τοῖς τῆς ἀποτομῆς ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, τὸ χωρίον δυναμένη ῥητή ἐστιν.
5

περιεχέσθω γὰρ χωρίον τὸ ὑπὸ τῶν ΑΒ, ΓΔ ὑπὸ ἀποτομῆς τῆς ΑΒ καὶ τῆς ἐκ δύο ὀνομάτων τῆς ΓΔ, ἧς μεῖζον ὄνομα ἔστω τὸ ΓΕ, καὶ ἔστω τὰ ὀνόματα τῆς ἐκ δύο ὀνομάτων τὰ ΓΕ, ΕΔ σύμμετρά τε τοῖς τῆς ἀποτομῆς ὀνόμασι
10τοῖς ΑΖ, ΖΒ καὶ ἐν τῷ αὐτῷ λόγῳ, καὶ ἔστω τὸ ὑπὸ τῶν ΑΒ, ΓΔ δυναμένη Η: λέγω, ὅτι ῥητή ἐστιν Η.

Ἐκκείσθω γὰρ ῥητὴ Θ, καὶ τῷ
15ἀπὸ τῆς Θ ἴσον παρὰ τὴν ΓΔ παραβεβλήσθω πλάτος ποιοῦν τὴν ΚΛ: ἀποτομὴ ἄρα ἐστὶν ΚΛ, ἧς τὰ ὀνόματα ἔστω τὰ ΚΜ, ΜΛ σύμμετρα τοῖς τῆς ἐκ δύο ὀνομάτων ὀνόμασι τοῖς ΓΕ, ΕΔ καὶ ἐν τῷ αὐτῷ λόγῳ. ἀλλὰ καὶ αἱ ΓΕ, ΕΔ σύμμετροί τέ εἰσι ταῖς ΑΖ, ΖΒ καὶ ἐν τῷ αὐτῷ λόγῳ:
20ἔστιν ἄρα ὡς ΑΖ πρὸς τὴν ΖΒ, οὕτως ΚΜ πρὸς ΜΛ. ἐναλλὰξ ἄρα ἐστὶν ὡς ΑΖ πρὸς τὴν ΚΜ, οὕτως ΒΖ πρὸς τὴν ΛΜ: καὶ λοιπὴ ἄρα ΑΒ πρὸς λοιπὴν τὴν ΚΛ ἐστιν ὡς ΑΖ πρὸς ΚΜ. σύμμετρος δὲ ΑΖ τῇ ΚΜ: σύμμετρος ἄρα ἐστὶ καὶ ΑΒ τῇ ΚΛ. καί ἐστιν ὡς ΑΒ
25πρὸς ΚΛ, οὕτως τὸ ὑπὸ τῶν ΓΔ, ΑΒ πρὸς τὸ ὑπὸ τῶν ΓΔ, ΚΛ: σύμμετρον ἄρα ἐστὶ καὶ τὸ ὑπὸ τῶν ΓΔ, ΑΒ τῷ ὑπὸ τῶν ΓΔ, ΚΛ. ἴσον δὲ τὸ ὑπὸ τῶν ΓΔ, ΚΛ τῷ ἀπὸ τῆς Θ: σύμμετρον ἄρα ἐστὶ τὸ ὑπὸ τῶν ΓΔ, ΑΒ τῷ ἀπὸ τῆς Θ. τῷ δὲ ὑπὸ τῶν ΓΔ, ΑΒ ἴσον ἐστὶ τὸ ἀπὸ
30τῆς Η: σύμμετρον ἄρα ἐστὶ τὸ ἀπὸ τῆς Η τῷ ἀπὸ τῆς Θ. ῥητὸν δὲ τὸ ἀπὸ τῆς Θ: ῥητὸν ἄρα ἐστὶ καὶ τὸ ἀπὸ τῆς Η: ῥητὴ ἄρα ἐστὶν Η. καὶ δύναται τὸ ὑπὸ τῶν ΓΔ, ΑΒ.

ἐὰν ἄρα χωρίον περιέχηται ὑπὸ ἀποτομῆς καὶ τῆς ἐκ δύο ὀνομάτων, ἧς τὰ ὀνόματα σύμμετρά ἐστι τοῖς τῆς ἀποτομῆς
35ὀνόμασι καὶ ἐν τῷ αὐτῷ λόγῳ, τὸ χωρίον δυναμένη ῥητή ἐστιν.

Πόρισμα

καὶ γέγονεν ἡμῖν καὶ διὰ τούτου φανερόν, ὅτι δυνατόν ἐστι ῥητὸν χωρίον ὑπὸ ἀλόγων εὐθειῶν περιέχεσθαι.
40ὅπερ ἔδει δεῖξαι.

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