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ἐὰν κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τοῦ κέντρου ἐπὶ τὴν ἁφὴν ἐπιζευχθῇ τις εὐθεῖα, ἐπιζευχθεῖσα κάθετος ἔσται ἐπὶ τὴν ἐφαπτομένην.

κύκλου γὰρ τοῦ ΑΒΓ ἐφαπτέσθω τις εὐθεῖα ΔΕ
5κατὰ τὸ Γ σημεῖον, καὶ εἰλήφθω τὸ κέντρον τοῦ ΑΒΓ κύκλου τὸ Ζ, καὶ ἀπὸ τοῦ Ζ ἐπὶ τὸ Γ ἐπεζεύχθω ΖΓ: λέγω, ὅτι ΖΓ κάθετός ἐστιν ἐπὶ τὴν ΔΕ.

εἰ γὰρ μή, ἤχθω ἀπὸ τοῦ Ζ ἐπὶ τὴν ΔΕ κάθετος ΖΗ.

ἐπεὶ οὖν ὑπὸ ΖΗΓ γωνία ὀρθή ἐστιν, ὀξεῖα ἄρα
10ἐστὶν ὑπὸ ΖΓΗ: ὑπὸ δὲ τὴν μείζονα γωνίαν μείζων πλευρὰ ὑποτείνει: μείζων ἄρα ΖΓ τῆς ΖΗ: ἴση δὲ ΖΓ τῇ ΖΒ: μείζων ἄρα καὶ ΖΒ τῆς ΖΗ ἐλάττων τῆς μείζονος: ὅπερ ἐστὶν
15ἀδύνατον. οὐκ ἄρα ΖΗ κάθετός ἐστιν ἐπὶ τὴν ΔΕ. ὁμοίως δὴ δείξομεν, ὅτι οὐδ᾽ ἄλλη τις πλὴν τῆς ΖΓ: ΖΓ ἄρα κάθετός ἐστιν ἐπὶ τὴν ΔΕ.
20

ἐὰν ἄρα κύκλου ἐφάπτηταί τις εὐθεῖα, ἀπὸ δὲ τοῦ κέντρου ἐπὶ τὴν ἁφὴν ἐπιζευχθῇ τις εὐθεῖα, ἐπιζευχθεῖσα
20κάθετος ἔσται ἐπὶ τὴν ἐφαπτομένην: ὅπερ ἔδει δεῖξαι.

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