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εἰς τὸν δοθέντα κύκλον τετράγωνον ἐγγράψαι.

ἔστω δοθεὶς κύκλος ΑΒΓΔ: δεῖ δὴ εἰς τὸν ΑΒΓΔ κύκλον τετράγωνον ἐγγράψαι.

ἤχθωσαν τοῦ ΑΒΓΔ κύκλου δύο διάμετροι πρὸς
5ὀρθὰς ἀλλήλαις αἱ ΑΓ, ΒΔ, καὶ ἐπεζεύχθωσαν αἱ ΑΒ, ΒΓ, ΓΔ, ΔΑ.

καὶ ἐπεὶ ἴση ἐστὶν ΒΕ τῇ ΕΔ: κέντρον γὰρ τὸ Ε: κοινὴ δὲ καὶ πρὸς ὀρθὰς ΕΑ, βάσις ἄρα ΑΒ βάσει τῇ ΑΔ ἴση ἐστίν. διὰ τὰ αὐτὰ δὴ καὶ
10ἑκατέρα τῶν ΒΓ, ΓΔ ἑκατέρᾳ τῶν ΑΒ, ΑΔ ἴση ἐστίν: ἰσόπλευρον ἄρα ἐστὶ τὸ ΑΒΓΔ τετράπλευρον. λέγω δή, ὅτι καὶ ὀρθογώνιον. ἐπεὶ γὰρ ΒΔ εὐθεῖα διάμετρός ἐστι τοῦ ΑΒΓΔ κύκλου, ἡμικύκλιον ἄρα
15ἐστὶ τὸ ΒΑΔ: ὀρθὴ ἄρα ὑπὸ ΒΑΔ γωνία. διὰ τὰ αὐτὰ δὴ καὶ ἑκάστη τῶν ὑπὸ ΑΒΓ, ΒΓΔ, ΓΔΑ ὀρθή ἐστιν: ὀρθογώνιον ἄρα ἐστὶ τὸ ΑΒΓΔ τετράπλευρον. ἐδείχθη δὲ καὶ ἰσόπλευρον: τετράγωνον ἄρα ἐστίν. καὶ ἐγγέγραπται εἰς τὸν ΑΒΓΔ κύκλον.

εἰς ἄρα τὸν δοθέντα κύκλον τετράγωνον ἐγγέγραπται τὸ ΑΒΓΔ: ὅπερ ἔδει ποιῆσαι.

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