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ἐν κύκλῳ αἱ ἴσαι εὐθεῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι ἀλλήλαις εἰσίν.

ἔστω κύκλος ΑΒΓΔ, καὶ ἐν αὐτῷ ἴσαι εὐθεῖαι ἔστωσαν
5αἱ ΑΒ, ΓΔ: λέγω, ὅτι αἱ ΑΒ, ΓΔ ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου.

εἰλήφθω γὰρ τὸ κέντρον τοῦ ΑΒΓΔ κύκλου καὶ ἔστω τὸ Ε, καὶ ἀπὸ τοῦ Ε ἐπὶ τὰς ΑΒ, ΓΔ κάθετοι ἤχθωσαν αἱ ΕΖ, ΕΗ, καὶ ἐπεζεύχθωσαν αἱ ΑΕ, ΕΓ.
10

ἐπεὶ οὖν εὐθεῖά τις διὰ τοῦ κέντρου ΕΖ εὐθεῖάν τινα μὴ διὰ τοῦ κέντρου τὴν ΑΒ πρὸς ὀρθὰς τέμνει, καὶ δίχα αὐτὴν τέμνει. ἴση ἄρα ΑΖ τῇ ΖΒ: διπλῆ ἄρα ΑΒ τῆς ΑΖ. διὰ τὰ αὐτὰ δὴ καὶ ΓΔ τῆς ΓΗ ἐστι διπλῆ: καί
15ἐστιν ἴση ΑΒ τῇ ΓΔ: ἴση ἄρα καὶ ΑΖ τῇ ΓΗ. καὶ ἐπεὶ ἴση ἐστὶν ΑΕ τῇ ΕΓ, ἴσον καὶ τὸ ἀπὸ τῆς ΑΕ τῷ ἀπὸ τῆς ΕΓ. ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΕ ἴσα τὰ ἀπὸ τῶν ΑΖ, ΕΖ: ὀρθὴ γὰρ
20πρὸς τῷ Ζ γωνία: τῷ δὲ ἀπὸ τῆς ΕΓ ἴσα τὰ ἀπὸ τῶν ΕΗ, ΗΓ: ὀρθὴ γὰρ πρὸς τῷ Η γωνία: τὰ ἄρα ἀπὸ τῶν ΑΖ, ΖΕ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΓΗ, ΗΕ, ὧν τὸ ἀπὸ τῆς ΑΖ ἴσον ἐστὶ τῷ ἀπὸ τῆς ΓΗ: ἴση γάρ ἐστιν ΑΖ τῇ ΓΗ: λοιπὸν ἄρα τὸ ἀπὸ τῆς ΖΕ τῷ ἀπὸ τῆς ΕΗ ἴσον
25ἐστίν: ἴση ἄρα ΕΖ τῇ ΕΗ. ἐν δὲ κύκλῳ ἴσον ἀπέχειν ἀπὸ τοῦ κέντρου εὐθεῖαι λέγονται, ὅταν αἱ ἀπὸ τοῦ κέντρου ἐπ᾽ αὐτὰς κάθετοι ἀγόμεναι ἴσαι ὦσιν: αἱ ἄρα ΑΒ, ΓΔ ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου.

ἀλλὰ δὴ αἱ ΑΒ, ΓΔ εὐθεῖαι ἴσον ἀπεχέτωσαν ἀπὸ τοῦ
30κέντρου, τουτέστιν ἴση ἔστω ΕΖ τῇ ΕΗ. λέγω, ὅτι ἴση ἐστὶ καὶ ΑΒ τῇ ΓΔ.

τῶν γὰρ αὐτῶν κατασκευασθέντων ὁμοίως δείξομεν, ὅτι διπλῆ ἐστιν μὲν ΑΒ τῆς ΑΖ, δὲ ΓΔ τῆς ΓΗ: καὶ ἐπεὶ ἴση ἐστὶν ΑΕ τῇ ΓΕ, ἴσον ἐστὶ τὸ ἀπὸ τῆς ΑΕ τῷ
35ἀπὸ τῆς ΓΕ: ἀλλὰ τῷ μὲν ἀπὸ τῆς ΑΕ ἴσα ἐστὶ τὰ ἀπὸ τῶν ΕΖ, ΖΑ, τῷ δὲ ἀπὸ τῆς ΓΕ ἴσα τὰ ἀπὸ τῶν ΕΗ,
35ΗΓ. τὰ ἄρα ἀπὸ τῶν ΕΖ, ΖΑ ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΕΗ, ΗΓ: ὧν τὸ ἀπὸ τῆς ΕΖ τῷ ἀπὸ τῆς ΕΗ ἐστιν ἴσον: ἴση γὰρ ΕΖ τῇ ΕΗ: λοιπὸν ἄρα τὸ ἀπὸ τῆς ΑΖ ἴσον ἐστὶ τῷ
40ἀπὸ τῆς ΓΗ: ἴση ἄρα ΑΖ τῇ ΓΗ: καί ἐστι τῆς μὲν ΑΖ διπλῆ ΑΒ, τῆς δὲ ΓΗ διπλῆ ΓΔ: ἴση ἄρα ΑΒ τῇ ΓΔ.

ἐν κύκλῳ ἄρα αἱ ἴσαι εὐθεῖαι ἴσον ἀπέχουσιν ἀπὸ τοῦ κέντρου, καὶ αἱ ἴσον ἀπέχουσαι ἀπὸ τοῦ κέντρου ἴσαι
45ἀλλήλαις εἰσίν: ὅπερ ἔδει δεῖξαι.

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