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ἐὰν τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν.
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τριγώνου γὰρ τοῦ ΑΒΓ τὸ ἀπὸ μιᾶς τῆς ΒΓ πλευρᾶς τετράγωνον ἴσον ἔστω τοῖς ἀπὸ τῶν ΒΑ, ΑΓ πλευρῶν τετραγώνοις: λέγω, ὅτι ὀρθή ἐστιν ὑπὸ ΒΑΓ γωνία.

ἤχθω γὰρ ἀπὸ τοῦ Α σημείου τῇ ΑΓ εὐθείᾳ πρὸς ὀρθὰς ΑΔ καὶ κείσθω τῇ ΒΑ ἴση ΑΔ, καὶ ἐπεζεύχθω
10 ΔΓ. ἐπεὶ ἴση ἐστὶν ΔΑ τῇ ΑΒ, ἴσον ἐστὶ καὶ τὸ ἀπὸ τῆς ΔΑ τετράγωνον τῷ ἀπὸ τῆς ΑΒ τετραγώνῳ. κοινὸν προσκείσθω τὸ ἀπὸ τῆς ΑΓ τετράγωνον: τὰ ἄρα ἀπὸ τῶν ΔΑ, ΑΓ τετράγωνα ἴσα ἐστὶ τοῖς ἀπὸ τῶν ΒΑ, ΑΓ τετραγώνοις. ἀλλὰ τοῖς μὲν ἀπὸ
15τῶν ΔΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ τῆς ΔΓ: ὀρθὴ γάρ ἐστιν ὑπὸ ΔΑΓ γωνία: τοῖς δὲ ἀπὸ τῶν ΒΑ, ΑΓ ἴσον ἐστὶ τὸ ἀπὸ ΒΓ: ὑπόκειται γάρ: τὸ ἄρα ἀπὸ τῆς ΔΓ τετράγωνον ἴσον ἐστὶ τῷ ἀπὸ τῆς ΒΓ τετραγώνῳ: ὥστε καὶ
20πλευρὰ ΔΓ τῇ ΒΓ ἐστιν ἴση: καὶ ἐπεὶ ἴση ἐστὶν ΔΑ τῇ ΑΒ, κοινὴ δὲ ΑΓ, δύο δὴ αἱ ΔΑ, ΑΓ δύο ταῖς ΒΑ, ΑΓ ἴσαι εἰσίν: καὶ βάσις ΔΓ βάσει τῇ ΒΓ ἴση: γωνία ἄρα ὑπὸ ΔΑΓ γωνίᾳ τῇ ὑπὸ ΒΑΓ ἐστιν ἴση. ὀρθὴ δὲ ὑπὸ ΔΑΓ: ὀρθὴ ἄρα καὶ
25ὑπὸ ΒΑΓ.

ἐὰν ἄρα τριγώνου τὸ ἀπὸ μιᾶς τῶν πλευρῶν τετράγωνον ἴσον τοῖς ἀπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν τετραγώνοις, περιεχομένη γωνία ὑπὸ τῶν λοιπῶν τοῦ τριγώνου δύο πλευρῶν ὀρθή ἐστιν: ὅπερ ἔδει δεῖξαι.

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