Pappus Alexandrinus, Mathematical Collection, Book 8, 1876

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    <archimedes>
      <text>
        <body>
          <chap>
            <p>
              <s id="id.000047">
                <pb n="1038"/>
              πρὸς ΖΕ, οὕτως ἡ ΗΖ πρὸς ΖΛ, διπλῆ δὲ ἡ ΒΖ τῆς
                <lb n="1"/>
              ΖΕ, διπλῆ ἄρα καὶ ἡ ΗΖ τῆς ΖΛ. </s>
              <s id="id.000048">τριγώνου δὴ τοῦ ΗΘΚ
                <lb n="2"/>
              διχοτομία ἡ ΗΛ, καὶ διπλῆ ἡ ΗΖ τῆς ΖΛ· τὸ Ζ ἄρα κέν-
                <lb n="3"/>
              τρον βάρους ἐστὶν τοῦ ΗΘΚ τριγώνου. </s>
              <s id="id.000049">ἦν δὲ καὶ τοῦ ΑΒΓ.
                <lb n="4"/>
              </s>
            </p>
            <p>
              <s id="id.000050">δ#. </s>
              <s id="id.000051">Τὸ δὲ ὑπερτεθὲν νῦν δειχθήσεται. </s>
              <s id="id.000052">ἔστω γὰρ ὡς
                <lb n="5"/>
              ἡ ΓΔ πρὸς ΔΘ, ἡ ΓΕ πρὸς ΕΚ, καὶ ἐπεζεύχθωσαν αἱ
                <lb n="6"/>
              ΔΕ ΘΚ τέμνουσαι ἀλλήλας κατὰ τὸ Λ· ὅτι ἴση μέν ἐστιν
                <lb n="7"/>
              ἡ ΘΛ τῇ ΚΛ, ὁ δὲ τῆς ΔΛ πρὸς ΛΕ λόγος σύγκειται
                <lb n="8"/>
              ἔκ τε τοῦ τῆς ΔΘ πρὸς ΘΓ καὶ τοῦ τῆς ΓΚ πρὸς ΚΕ.
                <lb n="9"/>
              </s>
            </p>
            <p>
              <s id="id.000053">Ἤχθω διὰ τοῦ Γ τῇ ΘΚ παράλληλος ἡ ΓΖ καὶ συμ-
                <lb n="10"/>
              πιπτέτω τῇ ΔΕ ἐκβληθείσῃ κατὰ τὸ Ζ. </s>
              <s id="id.000054">ἐπεὶ οὖν δύο εὐ-
                <lb n="11"/>
              θεῖαί εἰσιν αἱ ΔΛ ΛΕ, καὶ ἔξωθεν ἡ ΖΛ, ὁ ἄρα τῆς ΔΛ
                <lb n="12"/>
              πρὸς ΛΕ λόγος σύγκειται ἔκ τε τοῦ τῆς ΔΛ πρὸς ΛΖ καὶ
                <lb n="13"/>
              τοῦ τῆς ΛΖ πρὸς ΕΛ. </s>
              <s id="id.000055">ἀλλὰ τῷ μὲν τῆς ΔΛ πρὸς ΛΖ
                <lb n="14"/>
              λόγῳ ὁ αὐτός ἐστιν ὁ τῆς ΔΘ πρὸς ΘΓ διὰ τὸ παράλλη-
                <lb n="15"/>
              λον εἶναι τὴν ΓΖ τῇ ΚΘ, τῷ δὲ τῆς ΖΛ πρὸς ΛΕ λόγῳ
                <lb n="16"/>
              ὁ αὐτός ἐστιν ὁ τῆς ΓΚ πρὸς ΚΕ διὰ τὸ ἰσογώνια εἶναι
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              τὰ ΓΕΖ ΕΚΛ τρίγωνα· καὶ ὁ τῆς ΔΛ ἄρα πρὸς τὴν ΛΕ
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              λόγος σύγκειται ἔκ τε τοῦ τῆς ΔΘ πρὸς ΘΓ καὶ ἐκ τοῦ
                <lb n="19"/>
              τῆς ΓΚ πρὸς ΚΕ. </s>
              <s id="id.000056">κατὰ ταὐτὰ δὴ δειχθήσεται ὅτι καὶ ὁ
                <lb n="20"/>
              τῆς ΚΛ πρὸς ΛΘ λόγος συνῆπται ἔκ τε τοῦ τῆς ΚΕ πρὸς
                <lb n="21"/>
              ΕΓ καὶ τοῦ τῆς ΓΔ πρὸς ΔΘ, παραλλήλου ἀχθείσης τῇ
                <lb n="22"/>
              ΕΔ διὰ τοῦ Γ τῆς ΓΜ καὶ συμπιπτούσης τῇ ΚΘ ἐκβλη-
                <lb n="23"/>
              θείσῃ κατὰ τὸ Μ. </s>
              <s id="id.000057">ἐπεὶ γὰρ πάλιν δύο εὐθεῖαί εἰσιν αἱ
                <lb n="24"/>
              ΚΛ ΛΘ ἔξωθεν τῆς ΛΜ λαμβανομένης, ὁ ἄρα τῆς ΚΛ
                <lb n="25"/>
              πρὸς ΛΘ λόγος σύγκειται ἔκ τε τοῦ τῆς ΚΛ πρὸς ΛΜ καὶ τοῦ
                <lb n="26"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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