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ἐὰν ἐν κύκλῳ δύο εὐθεῖαι τέμνωσιν ἀλλήλας μὴ διὰ τοῦ κέντρου οὖσαι, οὐ τέμνουσιν ἀλλήλας δίχα.

ἔστω κύκλος ΑΒΓΔ, καὶ ἐν αὐτῷ δύο εὐθεῖαι αἱ ΑΓ, ΒΔ τεμνέτωσαν ἀλλήλας
5κατὰ τὸ Ε μὴ διὰ τοῦ κέντρου οὖσαι: λέγω, ὅτι οὐ τέμνουσιν ἀλλήλας δίχα.

εἰ γὰρ δυνατόν, τεμνέτωσαν ἀλλήλας δίχα ὥστε ἴσην εἶναι τὴν μὲν ΑΕ τῇ ΕΓ, τὴν δὲ ΒΕ τῇ ΕΔ: καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓΔ
10κύκλου, καὶ ἔστω τὸ Ζ, καὶ ἐπεζεύχθω ΖΕ.

ἐπεὶ οὖν εὐθεῖά τις διὰ τοῦ κέντρου ΖΕ εὐθεῖάν τινα μὴ διὰ τοῦ κέντρου τὴν ΑΓ δίχα τέμνει, καὶ πρὸς ὀρθὰς αὐτὴν τέμνει: ὀρθὴ ἄρα ἐστὶν ὑπὸ ΖΕΑ: πάλιν, ἐπεὶ εὐθεῖά τις ΖΕ εὐθεῖάν τινα τὴν ΒΔ δίχα τέμνει, καὶ
15πρὸς ὀρθὰς αὐτὴν τέμνει: ὀρθὴ ἄρα ὑπὸ ΖΕΒ. ἐδείχθη δὲ καὶ ὑπὸ ΖΕΑ ὀρθή: ἴση ἄρα ὑπὸ ΖΕΑ τῇ ὑπὸ ΖΕΒ ἐλάττων τῇ μείζονι: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα αἱ ΑΓ, ΒΔ τέμνουσιν ἀλλήλας δίχα.

ἐὰν ἄρα ἐν κύκλῳ δύο εὐθεῖαι τέμνωσιν ἀλλήλας μὴ διὰ
20τοῦ κέντρου οὖσαι, οὐ τέμνουσιν ἀλλήλας δίχα: ὅπερ ἔδει δεῖξαι.

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load focus English (Thomas L. Heath, Sir Thomas Little Heath, 1956)
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