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ἐν κύκλῳ πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ τῆς πρὸς τῇ περιφερείᾳ, ὅταν τὴν αὐτὴν περιφέρειαν βάσιν ἔχωσιν αἱ γωνίαι.

ἔστω κύκλος ΑΒΓ, καὶ πρὸς μὲν τῷ κέντρῳ αὐτοῦ
5γωνία ἔστω ὑπὸ ΒΕΓ, πρὸς δὲ τῇ περιφερείᾳ ὑπὸ ΒΑΓ, ἐχέτωσαν δὲ τὴν αὐτὴν περιφέρειαν βάσιν τὴν ΒΓ: λέγω, ὅτι διπλασίων ἐστὶν ὑπὸ ΒΕΓ γωνία τῆς ὑπὸ ΒΑΓ.

ἐπιζευχθεῖσα γὰρ ΑΕ διήχθω ἐπὶ τὸ Ζ.

ἐπεὶ οὖν ἴση ἐστὶν ΕΑ τῇ ΕΒ, ἴση καὶ γωνία ὑπὸ ΕΑΒ τῇ ὑπὸ ΕΒΑ: αἱ ἄρα ὑπὸ ΕΑΒ, ΕΒΑ γωνίαι τῆς ὑπὸ ΕΑΒ διπλασίους εἰσίν. ἴση δὲ ὑπὸ ΒΕΖ ταῖς ὑπὸ ΕΑΒ, ΕΒΑ: καὶ ὑπὸ ΒΕΖ ἄρα τῆς ὑπὸ ΕΑΒ ἐστι διπλῆ. διὰ τὰ αὐτὰ δὴ καὶ ὑπὸ ΖΕΓ τῆς ὑπὸ ΕΑΓ
15ἐστι διπλῆ. ὅλη ἄρα ὑπὸ ΒΕΓ ὅλης τῆς ὑπὸ ΒΑΓ ἐστι διπλῆ.

Κεκλάσθω δὴ πάλιν, καὶ ἔστω ἑτέρα γωνία ὑπὸ ΒΔΓ, καὶ ἐπιζευχθεῖσα ΔΕ ἐκβεβλήσθω ἐπὶ τὸ Η. ὁμοίως δὴ
20δείξομεν, ὅτι διπλῆ ἐστιν ὑπὸ ΗΕΓ γωνία τῆς ὑπὸ ΕΔΓ, ὧν ὑπὸ ΗΕΒ διπλῆ ἐστι τῆς ὑπὸ ΕΔΒ: λοιπὴ ἄρα ὑπὸ ΒΕΓ διπλῆ ἐστι τῆς ὑπὸ ΒΔΓ.

ἐν κύκλῳ ἄρα πρὸς τῷ κέντρῳ γωνία διπλασίων ἐστὶ
25τῆς πρὸς τῇ περιφερείᾳ, ὅταν τὴν αὐτὴν περιφέρειαν βάσιν ἔχωσιν αἱ γωνίαι: ὅπερ ἔδει δεῖξαι.

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