Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div758" type="section" level="1" n="259">
          <p>
            <s xml:id="echoid-s12146" xml:space="preserve">
              <pb o="264" file="294" n="294" rhead="GEOMETR. PRACT."/>
            atque inter AB, AE, media proportionali A G; </s>
            <s xml:id="echoid-s12147" xml:space="preserve">ac deniqueinter AB, AF, media
              <lb/>
            proportionali AH; </s>
            <s xml:id="echoid-s12148" xml:space="preserve">ducantur EI, GK, HL, lateri BC, parallelæ@ quas dico trian-
              <lb/>
            gulum partiriin 4. </s>
            <s xml:id="echoid-s12149" xml:space="preserve">partes æquales. </s>
            <s xml:id="echoid-s12150" xml:space="preserve"> Quoniam enim triangulum ABC,
              <note symbol="a" position="left" xlink:label="note-294-01" xlink:href="note-294-01a" xml:space="preserve">coroll. 4.
                <lb/>
              ſexti.</note>
            lo AEI, ſimile eſt; </s>
            <s xml:id="echoid-s12151" xml:space="preserve"> erit triangulum ABC, ad triangulum AEI, vt A B, ad A D, quod tres A B, A E, A D, ſint continuè pro portionales. </s>
            <s xml:id="echoid-s12152" xml:space="preserve">Eſt autem A D, quarta
              <lb/>
              <note symbol="b" position="left" xlink:label="note-294-02" xlink:href="note-294-02a" xml:space="preserve">coroll. 19.
                <lb/>
              ſexti.</note>
            parsipſius AB. </s>
            <s xml:id="echoid-s12153" xml:space="preserve">Igitur & </s>
            <s xml:id="echoid-s12154" xml:space="preserve">triangulum AEI, quarta pars eſt trianguli ABC.</s>
            <s xml:id="echoid-s12155" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12156" xml:space="preserve">
              <emph style="sc">Non</emph>
            aliter oſtendemus, eſſetriangulum A B C, ad triangulum A G K,
              <note symbol="c" position="left" xlink:label="note-294-03" xlink:href="note-294-03a" xml:space="preserve">coroll. 19.
                <lb/>
              ſexti.</note>
            AB, ad AE, quod etiam tres AB, AG, AE, ſint continue proportionales. </s>
            <s xml:id="echoid-s12157" xml:space="preserve">Quare
              <lb/>
            cum AE, contineat {2/4}. </s>
            <s xml:id="echoid-s12158" xml:space="preserve">rectæ AB, continebit etiam AGK, triangulum {2/4}. </s>
            <s xml:id="echoid-s12159" xml:space="preserve">trian-
              <lb/>
            guli A B C: </s>
            <s xml:id="echoid-s12160" xml:space="preserve">Ideoq; </s>
            <s xml:id="echoid-s12161" xml:space="preserve">cum AEI, ſit {1/4}. </s>
            <s xml:id="echoid-s12162" xml:space="preserve">trianguli A B C, vt oſtendimus, erit EIK G, {1/4}.
              <lb/>
            </s>
            <s xml:id="echoid-s12163" xml:space="preserve">eiuſdem trianguli A B C. </s>
            <s xml:id="echoid-s12164" xml:space="preserve">Deniq; </s>
            <s xml:id="echoid-s12165" xml:space="preserve">eadem ratione erit triangulum A B C, ad trian-
              <lb/>
            gulum AHL, vt AB, ad AF, quod etiam tres AB, AH, AF, ſint continue propor-
              <lb/>
            tionales: </s>
            <s xml:id="echoid-s12166" xml:space="preserve">ac proinde triangulum AHL, complectetur {3/4}. </s>
            <s xml:id="echoid-s12167" xml:space="preserve">trianguli ABC; </s>
            <s xml:id="echoid-s12168" xml:space="preserve">quem-
              <lb/>
            admodum AF, continet {3/4}. </s>
            <s xml:id="echoid-s12169" xml:space="preserve">ipſius AB: </s>
            <s xml:id="echoid-s12170" xml:space="preserve">ideoq; </s>
            <s xml:id="echoid-s12171" xml:space="preserve">BHLC, erit {1/4}. </s>
            <s xml:id="echoid-s12172" xml:space="preserve">trianguli ABC, &</s>
            <s xml:id="echoid-s12173" xml:space="preserve">c.</s>
            <s xml:id="echoid-s12174" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div761" type="section" level="1" n="260">
          <head xml:id="echoid-head285" xml:space="preserve">PROBL. 7. PROPOS. 12.</head>
          <p>
            <s xml:id="echoid-s12175" xml:space="preserve">DATVM triangulum per rectam ex puncto extra triangulum dato
              <lb/>
            ductam in duas partes æquales diuidere.</s>
            <s xml:id="echoid-s12176" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s12177" xml:space="preserve">Ex puncto D, extra triangulum AB C, dato ducenda ſit linea diuidens tri-
              <lb/>
            angulum bifariam. </s>
            <s xml:id="echoid-s12178" xml:space="preserve">Ducta recta D A, ad angulum oppoſitum ſecante latus B C,
              <lb/>
            in E: </s>
            <s xml:id="echoid-s12179" xml:space="preserve">ſi quidem B C, in E, diuiditur bifariam, factum erit, quod diubetur: </s>
            <s xml:id="echoid-s12180" xml:space="preserve"> quod tunc triangula ABE, ACE, ſint æqualia. </s>
            <s xml:id="echoid-s12181" xml:space="preserve">Si vero B C, non bifariam diuiditur in
              <lb/>
              <note symbol="d" position="left" xlink:label="note-294-04" xlink:href="note-294-04a" xml:space="preserve">38. primi.</note>
            E, ſit ſegmentum CE, maius, cui ducatur parallelaDF, occurrens lateri AC, pro-
              <lb/>
            ducto in F. </s>
            <s xml:id="echoid-s12182" xml:space="preserve">Secto latere AC, bifariam in G, inueniatur tribus DF. </s>
            <s xml:id="echoid-s12183" xml:space="preserve">BC, CG, quar-
              <lb/>
            ta proportionalis CH; </s>
            <s xml:id="echoid-s12184" xml:space="preserve"> Eritque rectangulum ſub DF, CH, æquale rectangulo ſub B C, C G; </s>
            <s xml:id="echoid-s12185" xml:space="preserve">hoc eſt ſemiſsirectanguli ſub B C, C A: </s>
            <s xml:id="echoid-s12186" xml:space="preserve"> cum rectangulum
              <note symbol="e" position="left" xlink:label="note-294-05" xlink:href="note-294-05a" xml:space="preserve">16. ſexti.</note>
            B C, C A, duplum ſit rectanguli ſub B C, C G. </s>
            <s xml:id="echoid-s12187" xml:space="preserve">Deinde
              <lb/>
              <note symbol="f" position="left" xlink:label="note-294-06" xlink:href="note-294-06a" xml:space="preserve">1. ſexti.</note>
              <figure xlink:label="fig-294-01" xlink:href="fig-294-01a" number="199">
                <image file="294-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/294-01"/>
              </figure>
            inuenta L, media proportionaliinter F C, C H, vt qua- dratum ex L, æquale ſitrectangulo ſub FC, CH, adiũ-
              <lb/>
              <note symbol="g" position="left" xlink:label="note-294-07" xlink:href="note-294-07a" xml:space="preserve">17. ſexti.</note>
            gaturipſi CH, recta H@, vtrectangulum ſub tota CI, & </s>
            <s xml:id="echoid-s12188" xml:space="preserve">
              <lb/>
            adiuncta HI, ęquale ſit quadrato exL, ſiue rectangulo
              <lb/>
            ſub F C, C H, quemadmodum ad finem ſcholij pro-
              <lb/>
            poſ. </s>
            <s xml:id="echoid-s12189" xml:space="preserve">36. </s>
            <s xml:id="echoid-s12190" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s12191" xml:space="preserve">3. </s>
            <s xml:id="echoid-s12192" xml:space="preserve">Euclid. </s>
            <s xml:id="echoid-s12193" xml:space="preserve">ſcripſimus: </s>
            <s xml:id="echoid-s12194" xml:space="preserve">ducaturque recta DI,
              <lb/>
            ſecans BC, in K. </s>
            <s xml:id="echoid-s12195" xml:space="preserve">Dico rectam DI, ſecare triangulum ABC, in duas partes AB-
              <lb/>
            KI, IKC, æquales. </s>
            <s xml:id="echoid-s12196" xml:space="preserve">Quoniam enim per conſtructionem rectangulũ ſub CI, IH,
              <lb/>
            æquale eſt rectangulo ſub CF, CH, erit vt CI, ad CF, ita CH, ad IH; </s>
            <s xml:id="echoid-s12197" xml:space="preserve">Et cõuer- tendo, vt CF, ad CI, ita IH, ad CH: </s>
            <s xml:id="echoid-s12198" xml:space="preserve">& </s>
            <s xml:id="echoid-s12199" xml:space="preserve">cõponẽdo vt IF, ad CI, ad CH. </s>
            <s xml:id="echoid-s12200" xml:space="preserve">
              <note symbol="h" position="left" xlink:label="note-294-08" xlink:href="note-294-08a" xml:space="preserve">16. ſexti.</note>
            aũt IF, ad CI, ita eſt FD, ad CK. </s>
            <s xml:id="echoid-s12201" xml:space="preserve">Igitur erit quoq; </s>
            <s xml:id="echoid-s12202" xml:space="preserve">FD, ad CK, vt CI, ad CH: </s>
            <s xml:id="echoid-s12203" xml:space="preserve">
              <note symbol="i" position="left" xlink:label="note-294-09" xlink:href="note-294-09a" xml:space="preserve">4. ſexti &
                <lb/>
              permutando.</note>
            {pue}inde rectangulũ ſub FD, CH æquale erit rectangulo ſub CK, CI: </s>
            <s xml:id="echoid-s12204" xml:space="preserve">Erat aũtre-
              <lb/>
            ctangulũ ſub FD, CH, per conſtructionẽ æquale ſemiſsirectanguli ſub BC, CA.
              <lb/>
            </s>
            <s xml:id="echoid-s12205" xml:space="preserve">
              <note symbol="k" position="left" xlink:label="note-294-10" xlink:href="note-294-10a" xml:space="preserve">16. ſexti.</note>
            Igitur & </s>
            <s xml:id="echoid-s12206" xml:space="preserve">rectangulum ſub C K, C I, æquale erit ſemiſsi rectanguli ſub B C,
              <lb/>
            C A. </s>
            <s xml:id="echoid-s12207" xml:space="preserve">Vt autem rectangulum ſub CK, CI, ad rectangulum ſub BC, CA, ita eſt
              <note symbol="l" position="left" xlink:label="note-294-11" xlink:href="note-294-11a" xml:space="preserve">9. hui{us}.</note>
            angulum CKI, ad triangulum ABC. </s>
            <s xml:id="echoid-s12208" xml:space="preserve">Igitur triangulum CKI, æquale </s>
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