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μείζων κατὰ τὸ αὐτὸ μόνον σημεῖον διαιρεῖται.

ἔστω μείζων ΑΒ διῃρημένη κατὰ τὸ Γ, ὥστε τὰς ΑΓ, ΓΒ δυνάμει ἀσυμμέτρους εἶναι ποιούσας τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΓ, ΓΒ τετραγώνων ῥητόν,
5τὸ δ᾽ ὑπὸ τῶν ΑΓ, ΓΒ μέσον: λέγω, ὅτι ΑΒ κατ᾽ ἄλλο σημεῖον οὐ διαιρεῖται.

εἰ γὰρ δυνατόν, διῃρήσθω καὶ κατὰ τὸ Δ, ὥστε καὶ τὰς ΑΔ, ΔΒ δυνάμει ἀσυμμέτρους εἶναι ποιούσας τὸ μὲν συγκείμενον ἐκ τῶν ἀπὸ τῶν ΑΔ, ΔΒ
10ῥητόν, τὸ δ᾽ ὑπ᾽ αὐτῶν μέσον. καὶ ἐπεί, διαφέρει τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ, τούτῳ διαφέρει καὶ τὸ δὶς ὑπὸ τῶν ΑΔ, ΔΒ τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ, ἀλλὰ τὰ ἀπὸ τῶν ΑΓ, ΓΒ τῶν ἀπὸ τῶν ΑΔ, ΔΒ ὑπερέχει ῥητῷ: ῥητὰ γὰρ ἀμφότερα: καὶ τὸ δὶς ὑπὸ τῶν
15ΑΔ, ΔΒ ἄρα τοῦ δὶς ὑπὸ τῶν ΑΓ, ΓΒ ὑπερέχει ῥητῷ μέσα ὄντα: ὅπερ ἐστὶν ἀδύνατον. οὐκ ἄρα μείζων κατ᾽ ἄλλο καὶ ἄλλο σημεῖον διαιρεῖται: κατὰ τὸ αὐτὸ ἄρα μόνον διαιρεῖται: ὅπερ ἔδει δεῖξαι.

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