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εἰς τὸν δοθέντα κύκλον τῷ δοθέντι τριγώνῳ ἰσογώνιον τρίγωνον ἐγγράψαι.

ἔστω δοθεὶς κύκλος ΑΒΓ, τὸ δὲ δοθὲν τρίγωνον τὸ ΔΕΖ: δεῖ δὴ εἰς τὸν ΑΒΓ κύκλον τῷ ΔΕΖ τριγώνῳ
5ἰσογώνιον τρίγωνον ἐγγράψαι.

ἤχθω τοῦ ΑΒΓ κύκλου ἐφαπτομένη ΗΘ κατὰ τὸ Α, καὶ συνεστάτω πρὸς τῇ ΑΘ εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α τῇ ὑπὸ ΔΕΖ γωνίᾳ ἴση ὑπὸ ΘΑΓ, πρὸς δὲ τῇ ΑΗ
10εὐθείᾳ καὶ τῷ πρὸς αὐτῇ σημείῳ τῷ Α τῇ ὑπὸ ΔΖΕ γωνίᾳ ἴση ὑπὸ ΗΑΒ, καὶ ἐπεζεύχθω ΒΓ.

ἐπεὶ οὖν κύκλου τοῦ ΑΒΓ ἐφάπτεταί τις εὐθεῖα ΑΘ, καὶ
15ἀπὸ τῆς κατὰ τὸ Α ἐπαφῆς εἰς τὸν κύκλον διῆκται εὐθεῖα ΑΓ, ἄρα ὑπὸ ΘΑΓ ἴση ἐστὶ τῇ ἐν τῷ ἐναλλὰξ τοῦ κύκλου τμήματι γωνίᾳ τῇ ὑπὸ ΑΒΓ. ἀλλ᾽ ὑπὸ ΘΑΓ τῇ ὑπὸ ΔΕΖ ἐστιν ἴση: καὶ ὑπὸ ΑΒΓ ἄρα γωνία τῇ ὑπὸ ΔΕΖ ἐστιν ἴση. διὰ τὰ αὐτὰ δὴ καὶ ὑπὸ ΑΓΒ
20τῇ ὑπὸ ΔΖΕ ἐστιν ἴση: καὶ λοιπὴ ἄρα ὑπὸ ΒΑΓ λοιπῇ τῇ ὑπὸ ΕΔΖ ἐστιν ἴση: ἰσογώνιον ἄρα ἐστὶ τὸ ΑΒΓ τρίγωνον τῷ ΔΕΖ τριγώνῳ, καὶ ἐγγέγραπται εἰς τὸν ΑΒΓ κύκλον.

εἰς τὸν δοθέντα ἄρα κύκλον τῷ δοθέντι τριγώνῳ ἰσογώνιον
25τρίγωνον ἐγγέγραπται: ὅπερ ἔδει ποιῆσαι.

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