Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div688" type="section" level="1" n="243">
          <pb o="246" file="276" n="276" rhead="GEOMETR. PRACT."/>
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        <div xml:id="echoid-div699" type="section" level="1" n="244">
          <head xml:id="echoid-head269" xml:space="preserve">PROBL. 2. PROPOS. 3.</head>
          <p>
            <s xml:id="echoid-s11446" xml:space="preserve">DIVISO rectilineo quolibet in triangula ex vno aliquo puncto, rectas
              <lb/>
            lineas ipſis triangulis ordine proportionales inuenire.</s>
            <s xml:id="echoid-s11447" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s11448" xml:space="preserve">
              <emph style="sc">Sit</emph>
            rectilineum quo dlibet A B C D E F, diuiſum in triangula A B C, A C D,
              <lb/>
            ADE, AEF, per rectas ex angulo A, (vel aliquo puncto aſsignato in vno latere)
              <lb/>
            ad omnes angulos oppoſitos ductas: </s>
            <s xml:id="echoid-s11449" xml:space="preserve">atque hiſce triangulis inueniendæ ſint or-
              <lb/>
            dine totidem rectæ proportionales. </s>
            <s xml:id="echoid-s11450" xml:space="preserve">Ex omnibus angulis dempto angulo A, ad
              <lb/>
              <figure xlink:label="fig-276-01" xlink:href="fig-276-01a" number="179">
                <image file="276-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/276-01"/>
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            rectas ex A, egredientes ducantur perpendiculares B I,
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            CL, DK, DN, EM, FO, pro altitudinibus triangulorũ.
              <lb/>
            </s>
            <s xml:id="echoid-s11451" xml:space="preserve">(Nihil autem refert, ſi interdum perpendiculares cadant
              <lb/>
            in rectas extra triangula productas, cuiuſmodi hic ſunt
              <lb/>
            DK, DN,) ita vt ſingula triangula binas habeant altitudi-
              <lb/>
            nes, præter duo extrema, quæ ſingulas duntaxat habent. </s>
            <s xml:id="echoid-s11452" xml:space="preserve">
              <lb/>
            Deinde in recta quacunque GH, accipiatur GP, æ qualis
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            altitudini BI, primi trianguli ABC; </s>
            <s xml:id="echoid-s11453" xml:space="preserve">& </s>
            <s xml:id="echoid-s11454" xml:space="preserve">P Q, æqualis altitu-
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            dini DK, ſecundi trianguli A C D, reſpectu eiuſdem baſis
              <lb/>
            AC. </s>
            <s xml:id="echoid-s11455" xml:space="preserve">Poſt hæcſiat, vt CL altitudo
              <note symbol="a" position="left" xlink:label="note-276-01" xlink:href="note-276-01a" xml:space="preserve">12. ſexti.</note>
            ſpectu baſis AD, ad EM, altitudinẽ tertij trianguli ADE, reſpectu eiuſdem baſis
              <lb/>
            AD, ita PQ. </s>
            <s xml:id="echoid-s11456" xml:space="preserve">ad QR; </s>
            <s xml:id="echoid-s11457" xml:space="preserve">Et vt DN, altitudo tertij trianguli ADE, reſpectu baſis AE,
              <lb/>
            ad FO, altitudinem quartitrianguli AEF, reſpectu eiuſdem baſis AE, ita QR, ad
              <lb/>
            RH, atque ita deinceps, ſi plura fuerint triangula, ſumendo ſemper duas alti-
              <lb/>
            tudines ad communem baſem demiſſas, &</s>
            <s xml:id="echoid-s11458" xml:space="preserve">c. </s>
            <s xml:id="echoid-s11459" xml:space="preserve">Dico quatuor rectas G P, PQ,
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            QR, R H, eſſe quatuor triangulis ordine proportionales. </s>
            <s xml:id="echoid-s11460" xml:space="preserve">Nam vt in ſcholio
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            propoſ. </s>
            <s xml:id="echoid-s11461" xml:space="preserve">1. </s>
            <s xml:id="echoid-s11462" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s11463" xml:space="preserve">6. </s>
            <s xml:id="echoid-s11464" xml:space="preserve">Euclid demonſtratum eſt, à nobis, ita eſt triangulum A B C, ad
              <lb/>
            triangulum ACD, vt altitudo BI, ad altitudinem D K, propter baſem commu-
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            nem AC, hoc eſt, vt GP, ad PQ, cum hæ ſumptæ ſint illis altitudinibus æquales.
              <lb/>
            </s>
            <s xml:id="echoid-s11465" xml:space="preserve">Eadem de cauſa ita eſt triangulum ACD, ad triangulum ADE, vt altitudo CL,
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            ad altitudinem EM, hoc eſt, vt PQ. </s>
            <s xml:id="echoid-s11466" xml:space="preserve">ad QR, cum ex conſtructione ſit, vt CL, ad
              <lb/>
            EM, ita PQ. </s>
            <s xml:id="echoid-s11467" xml:space="preserve">ad QR. </s>
            <s xml:id="echoid-s11468" xml:space="preserve">Pari denique ratione ita eſt triangulum ADE, ad triangu-
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            lum AEF, vt altitudo DN, ad altitudinem, FO, hoc eſt, vt QR, ad R H, cum ſit
              <lb/>
            per conſtructionem, vt DM, ad FO, ita QR, ad RH. </s>
            <s xml:id="echoid-s11469" xml:space="preserve">Conſtat ergo id, quod pro-
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            poſitum fuit.</s>
            <s xml:id="echoid-s11470" xml:space="preserve"/>
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        <div xml:id="echoid-div701" type="section" level="1" n="245">
          <head xml:id="echoid-head270" xml:space="preserve">ALITER.</head>
          <p>
            <s xml:id="echoid-s11471" xml:space="preserve">
              <emph style="sc">Sit</emph>
            rurſus rectilineum ABCDEF, diuiſum in triangula ABC, ACD, ADE,
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            AEF, ex puncto A. </s>
            <s xml:id="echoid-s11472" xml:space="preserve">Quoniam bina proxima triangula conſtituunt qua drila-
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            terum, cuius diameter eſt latus vtrique triangulo commune, cuiuſmodi eſt A-
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            BCD, ducemus diametro AC, ex D, parallelam D O, quæ ſecet latus BC, pro-
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            ductum in O. </s>
            <s xml:id="echoid-s11473" xml:space="preserve">Sic in quadrilatero ACDE, diametro AD, parallelam ducemus
              <lb/>
            EP, quæ ſecet latus CD, protractum in P. </s>
            <s xml:id="echoid-s11474" xml:space="preserve">Itemque in quadrilatero ADEF, dia-
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            metro AE, parallelam ducemus FQ. </s>
            <s xml:id="echoid-s11475" xml:space="preserve">quæ latus DE, productum ſecet in Q. </s>
            <s xml:id="echoid-s11476" xml:space="preserve">De-
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            inde in recta quauis GN, ſumantur G H, HK, ipſis BC, CO, æquales, Et </s>
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