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τὴν δοθεῖσαν εá½Î¸Îµá¿–αν πεπεÏασμÎνην ἄκÏον καὶ μÎσον λόγον τεμεῖν. ἔστω ἡ δοθεῖσα εá½Î¸Îµá¿–α πεπεÏασμÎνη ἡ ΑΒ: δεῖ δὴ τὴν ΑΒ εá½Î¸Îµá¿–αν ἄκÏον καὶ μÎσον λόγον τεμεῖν.
5ἈναγεγÏάφθω ἀπὸ τῆς ΑΒ τετÏάγωνον τὸ ΒΓ, καὶ παÏαβεβλήσθω παÏá½° τὴν ΑΓ τῇ ΒΓ ἴσον παÏαλληλόγÏαμμον τὸ ΓΔ ὑπεÏβάλλον εἴδει Ï„á¿· ΑΔ á½Î¼Î¿Î¯á¿³ Ï„á¿· ΒΓ. τετÏάγωνον δΠá¼ÏƒÏ„ι τὸ ΒΓ: τετÏάγωνον ἄÏα á¼ÏƒÏ„ὶ καὶ
10τὸ ΑΔ. καὶ á¼Ï€Îµá½¶ ἴσον á¼ÏƒÏ„ὶ τὸ ΒΓ Ï„á¿· ΓΔ, κοινὸν ἀφῃÏήσθω τὸ ΓΕ: λοιπὸν ἄÏα τὸ ΒΖ λοιπῷ Ï„á¿· ΑΔ á¼ÏƒÏ„ιν ἴσον. ἔστι δὲ αá½Ï„á¿· καὶ ἰσογώνιον: τῶν ΒΖ, ΑΔ ἄÏα ἀντιπεπόνθασιν αἱ πλευÏαὶ αἱ πεÏὶ Ï„á½°Ï‚ ἴσας γωνίας: ἔστιν ἄÏα ὡς ἡ ΖΕ Ï€Ïὸς τὴν ΕΔ, οὕτως ἡ ΑΕ Ï€Ïὸς τὴν ΕΒ.
15ἴση δὲ ἡ μὲν ΖΕ τῇ ΑΒ, ἡ δὲ ΕΔ τῇ ΑΕ. ἔστιν ἄÏα ὡς ἡ ΒΑ Ï€Ïὸς τὴν ΑΕ, οὕτως ἡ ΑΕ Ï€Ïὸς τὴν ΕΒ. μείζων δὲ ἡ ΑΒ τῆς ΑΕ: μείζων ἄÏα καὶ ἡ ΑΕ τῆς ΕΒ. ἡ ἄÏα ΑΒ εá½Î¸Îµá¿–α ἄκÏον καὶ μÎσον λόγον Ï„Îτμηται κατὰ τὸ Ε, καὶ τὸ μεῖζον αá½Ï„ῆς τμῆμά á¼ÏƒÏ„ι τὸ ΑΕ:
20á½…Ï€ÎµÏ á¼”Î´ÎµÎ¹ ποιῆσαι.
Euclid. Euclidis Elementa. J. L. Heiberg. Leipzig. Teubner. 1883-1888.
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