1100ἡ ΑΖ, καὶ ἀπειλήφθω αὐτῆς τὸ γ# μέρος, καὶ ἔστω ἡ ΑΓ,
ἐφ' ἧς τμῆμα κύκλου γεγράφθω τὸ ΑΒΓ δεχόμενον γωνίαν
διμοίρου ὀρθῆς, καὶ οἵων ἐστὶν ἡ ΑΓ ε#, τοιούτων δ# ἀπει-
λήφθω ἡ ΓΕ, καὶ ἤχθω ἐφαπτομένη τοῦ τμήματος ἡ ΕΒ,
καὶ ἐπεζεύχθω ἥ τε ΑΒ καὶ ἡ ΖΒ, καὶ ἔτι ἐπιζευχθεῖσα
ἡ ΒΓ ἐκβεβλήσθω ἐπὶ τὸ Δ, καὶ κείσθω τῇ ΑΒ ἴση ἡ
ΒΔ, καὶ ἐπεζεύχθω ἡ ΑΔ. ἐπεὶ οὖν εἰς κύκλον διήχθησαν
ἥ τε ΕΓΑ καὶ ἡ ΕΒ, καὶ ἡ μὲν τέμνει τὸν κύκλον ἡ δὲ
ἐφάπτεται, τὸ ἄρα ὑπὸ ΑΕΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς ΕΒ·
ἔστιν ἄρα ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως ἡ ΒΕ πρὸς ΓΕ·
ἰσογώνιον ἄρα τὸ ΓΒΕ τρίγωνον τῷ ΑΒΕ τριγώνῳ. ἔστιν
ἄρα ὡς ἡ ΕΑ πρὸς ΑΒ, ἡ ΕΒ πρὸς ΒΓ· καὶ ὡς ἄρα τὸ
ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΕΒ, τὸ ἀπὸ τῆς ΑΒ πρὸς
τὸ ἀπὸ τῆς ΒΓ. ἀλλ' ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ
τῆς ΕΒ, οὕτως ἐστὶν ἡ ΑΕ πρὸς ΕΓ διὰ κ# τοῦ ς#. καὶ
ὡς ἄρα ἡ ΑΕ πρὸς ΕΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ, τουτέστιν
τὸ ἀπὸ τῆς ΒΔ, πρὸς τὸ ἀπὸ τῆς ΒΓ· τὸ ἄρα ἀπὸ τῆς
ΒΔ πρὸς τὸ ἀπὸ τῆς ΒΓ λόγον ἔχει ὃν τὰ θ# πρὸς δ#·
ἡμιολία ἄρα ἡ ΒΔ τῆς ΒΓ· διπλασία ἄρα ἡ ΒΓ τῆς ΓΔ.
ἔστιν δὲ καὶ ἡ ΖΓ τῆς ΓΑ διπλασία· ὡς ἄρα ἡ ΖΓ πρὸς
ΓΑ, ἡ ΒΓ πρὸς ΓΔ. καὶ ἴσαι εἰσὶν αἱ πρὸς τῷ Γ γω-
νίαι· ἴση ἄρα καὶ ἡ μὲν Δ γωνία τῇ ὑπὸ ΖΒΓ, ἡ δὲ Ζ
τῇ ὑπὸ ΓΑΔ· ἔστιν ἄρα ὡς ἡ ΖΒ πρὸς ΒΓ, οὕτως ἡ ΑΔ
πρὸς ΔΓ. ἐναλλὰξ ὡς ἡ ΖΒ πρὸς ΑΔ, οὕτως ἡ ΒΓ πρὸς
ΓΔ. διπλασία δὲ ἡ ΒΓ τῆς ΓΔ· διπλασία ἄρα καὶ ἡ ΖΒ
τῆς ΑΔ, τουτέστιν τῆς ΑΒ. καὶ ἔστιν διμοίρου ἡ Δ·
διμοίρου ἄρα ὀρθῆς καὶ ἡ ὑπὸ ΖΒΓ. ὅλη δὲ ἡ ὑπὸ ΑΒΖ
ἐφ' ἧς τμῆμα κύκλου γεγράφθω τὸ ΑΒΓ δεχόμενον γωνίαν
διμοίρου ὀρθῆς, καὶ οἵων ἐστὶν ἡ ΑΓ ε#, τοιούτων δ# ἀπει-
λήφθω ἡ ΓΕ, καὶ ἤχθω ἐφαπτομένη τοῦ τμήματος ἡ ΕΒ,
καὶ ἐπεζεύχθω ἥ τε ΑΒ καὶ ἡ ΖΒ, καὶ ἔτι ἐπιζευχθεῖσα
ἡ ΒΓ ἐκβεβλήσθω ἐπὶ τὸ Δ, καὶ κείσθω τῇ ΑΒ ἴση ἡ
ΒΔ, καὶ ἐπεζεύχθω ἡ ΑΔ. ἐπεὶ οὖν εἰς κύκλον διήχθησαν
ἥ τε ΕΓΑ καὶ ἡ ΕΒ, καὶ ἡ μὲν τέμνει τὸν κύκλον ἡ δὲ
ἐφάπτεται, τὸ ἄρα ὑπὸ ΑΕΓ ἴσον ἐστὶν τῷ ἀπὸ τῆς ΕΒ·
ἔστιν ἄρα ὡς ἡ ΑΕ πρὸς ΕΒ, οὕτως ἡ ΒΕ πρὸς ΓΕ·
ἰσογώνιον ἄρα τὸ ΓΒΕ τρίγωνον τῷ ΑΒΕ τριγώνῳ. ἔστιν
ἄρα ὡς ἡ ΕΑ πρὸς ΑΒ, ἡ ΕΒ πρὸς ΒΓ· καὶ ὡς ἄρα τὸ
ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ τῆς ΕΒ, τὸ ἀπὸ τῆς ΑΒ πρὸς
τὸ ἀπὸ τῆς ΒΓ. ἀλλ' ὡς τὸ ἀπὸ τῆς ΑΕ πρὸς τὸ ἀπὸ
τῆς ΕΒ, οὕτως ἐστὶν ἡ ΑΕ πρὸς ΕΓ διὰ κ# τοῦ ς#. καὶ
ὡς ἄρα ἡ ΑΕ πρὸς ΕΓ, οὕτως τὸ ἀπὸ τῆς ΑΒ, τουτέστιν
τὸ ἀπὸ τῆς ΒΔ, πρὸς τὸ ἀπὸ τῆς ΒΓ· τὸ ἄρα ἀπὸ τῆς
ΒΔ πρὸς τὸ ἀπὸ τῆς ΒΓ λόγον ἔχει ὃν τὰ θ# πρὸς δ#·
ἡμιολία ἄρα ἡ ΒΔ τῆς ΒΓ· διπλασία ἄρα ἡ ΒΓ τῆς ΓΔ.
ἔστιν δὲ καὶ ἡ ΖΓ τῆς ΓΑ διπλασία· ὡς ἄρα ἡ ΖΓ πρὸς
ΓΑ, ἡ ΒΓ πρὸς ΓΔ. καὶ ἴσαι εἰσὶν αἱ πρὸς τῷ Γ γω-
νίαι· ἴση ἄρα καὶ ἡ μὲν Δ γωνία τῇ ὑπὸ ΖΒΓ, ἡ δὲ Ζ
τῇ ὑπὸ ΓΑΔ· ἔστιν ἄρα ὡς ἡ ΖΒ πρὸς ΒΓ, οὕτως ἡ ΑΔ
πρὸς ΔΓ. ἐναλλὰξ ὡς ἡ ΖΒ πρὸς ΑΔ, οὕτως ἡ ΒΓ πρὸς
ΓΔ. διπλασία δὲ ἡ ΒΓ τῆς ΓΔ· διπλασία ἄρα καὶ ἡ ΖΒ
τῆς ΑΔ, τουτέστιν τῆς ΑΒ. καὶ ἔστιν διμοίρου ἡ Δ·
διμοίρου ἄρα ὀρθῆς καὶ ἡ ὑπὸ ΖΒΓ. ὅλη δὲ ἡ ὑπὸ ΑΒΖ