Loading ...
Sorry, an error occurred while loading the content.
 

Triangle construction given

Expand Messages
  • Luís Lopes
    Dear Hyacinthists, Source: an old russian geometry book translated to french. Hum.... hum.... hum?? How to put the data in a feasible position?? 2p(or s) =
    Message 1 of 9 , May 26, 2008
      Dear Hyacinthists,

      Source: an old russian geometry book translated
      to french.

      Hum.... hum.... hum??

      How to put the data in a feasible position??

      2p(or s) = a+b+c

      Best regards,
      Luis
      _________________________________________________________________
      Receba GRÁTIS as mensagens do Messenger no seu celular quando você estiver offline. Conheça o MSN Mobile!
      http://mobile.live.com/signup/signup2.aspx?lc=pt-br

      [Non-text portions of this message have been removed]
    • Francisco Javier García Capitán
      Luis, ¿qué significa feasible position in a construction problem? ... -- ... Francisco Javier García Capitán http://garciacapitan.auna.com [Non-text
      Message 2 of 9 , May 26, 2008
        Luis, ¿qué significa feasible position in a construction problem?

        2008/5/26, Luís Lopes <qed_texte@...>:
        >
        >
        > Dear Hyacinthists,
        >
        > Source: an old russian geometry book translated
        > to french.
        >
        > Hum.... hum.... hum??
        >
        > How to put the data in a feasible position??
        >
        > 2p(or s) = a+b+c
        >
        > Best regards,
        > Luis
        > __________________________________________________________
        > Receba GRÁTIS as mensagens do Messenger no seu celular quando você estiver
        > offline. Conheça o MSN Mobile!
        > http://mobile.live.com/signup/signup2.aspx?lc=pt-br
        >
        > [Non-text portions of this message have been removed]
        >
        >
        >



        --
        ---
        Francisco Javier García Capitán
        http://garciacapitan.auna.com


        [Non-text portions of this message have been removed]
      • Vladimir Dubrovsky
        Dear Luis! If you mark off segments BB =BA and CC =CA on the extensions of the side BC, you ll get a triangle AB C with the same h_a, B C =2p and
        Message 3 of 9 , May 26, 2008
          Dear Luis!

          If you mark off segments BB'=BA and CC'=CA on the extensions of the side BC,
          you'll get a triangle AB'C' with the same h_a, B'C'=2p and <B' - <C' = (<B -
          <C)/2. Then, consider the reflection C" of the point C' in the line parallel
          to BC and passing through A. We get an obviously constructible right
          triangle B'C'C" with <B'C'C"=90°, B'C'=2p, C'C"=2h_a.
          Then we know the distance h_a from point A to B'C'(=BC) and the angle B'AC"=
          <B' - <C' + 180° (as is easy to compute).So we can construct A and finish by
          constructing B and C at the intersections of B'C' with perpendicular
          bisectors to B'A and C'A.

          Friendly,
          Vladimir

          2008/5/26 Luís Lopes <qed_texte@...>:

          >
          > Dear Hyacinthists,
          >
          > Source: an old russian geometry book translated
          > to french.
          >
          > Hum.... hum.... hum??
          >
          > How to put the data in a feasible position??
          >
          > 2p(or s) = a+b+c
          >
          > Best regards,
          > Luis
          >


          [Non-text portions of this message have been removed]
        • Luís Lopes
          Dear Hyacinthists, [Francisco Capitan] Luis, ¿qué significa feasible position in a construction problem? Well, I tried to quote Petersen. My mistake, I
          Message 4 of 9 , May 26, 2008
            Dear Hyacinthists,

            [Francisco Capitan] Luis, ¿qué significa feasible position
            in a construction problem?
            Well, I tried to quote Petersen. My mistake, I should
            have written "suitable position".

            Dear Vladimir

            Thank you for your reply and solution.
            As always, they are very appreciated.

            Best regards,
            Luis

            _________________________________________________________________
            Instale a Barra de Ferramentas com Desktop Search e ganhe EMOTICONS para o Messenger! É GRÁTIS!
            http://www.msn.com.br/emoticonpack
          • Luís Lopes
            Dear Hyacinthists, Source: an old triangle construction german book. Its suggestion to both problems is to construct b+c. I don t figure out how. I think that
            Message 5 of 9 , May 26, 2008
              Dear Hyacinthists,

              Source: an old triangle construction german book.

              Its suggestion to both problems is to construct b+c.
              I don't figure out how.

              I think that here is more of a matter of knowing a
              previous result than to put the data in a suitable position.

              Best regards,
              Luis

              _________________________________________________________________
              Instale a Barra de Ferramentas com Desktop Search e ganhe EMOTICONS para o Messenger! É GRÁTIS!
              http://www.msn.com.br/emoticonpack
            • jpehrmfr
              Dear Luis ... First, b-c =(r_b-r_c)tan(A/2) In the first case, get L such as L^2 = (b-c)^2+4r^2/cos^2(A/2) then b+c= 4r/sin(A)+L In the second case, get L such
              Message 6 of 9 , May 26, 2008
                Dear Luis
                > Its suggestion to both problems is to construct b+c.
                > I don't figure out how.

                First, b-c =(r_b-r_c)tan(A/2)
                In the first case, get L such as
                L^2 = (b-c)^2+4r^2/cos^2(A/2)
                then b+c= 4r/sin(A)+L
                In the second case, get L such as
                L^2 = (b-c)^2+4r_a^2/cos^2(A/2)
                then b+c= 4r_a/sin(A)-L
                I recognize that this is not very nice but it works and I didn't find
                something better (although it's probably possible)
                Friendly
                Jean-Pierre
              • Vladimir Dubrovsky
                Dear Luis If you want just to construct the triangle, not necessarily using the sum b+c, then it suffices to notice that r_b-r_c=(b-c) cot A/2, so the problem
                Message 7 of 9 , May 26, 2008
                  Dear Luis

                  If you want just to construct the triangle, not necessarily using the sum
                  b+c, then it suffices to notice that
                  r_b-r_c=(b-c) cot A/2, so the problem reduces to the construction of a
                  triangle from A, r (or r_a) and b-c, which was discussed in post 16341 and
                  later.

                  Best regards,
                  Vladimir


                  2008/5/26 Luís Lopes <qed_texte@...>:

                  >
                  > Dear Hyacinthists,
                  >
                  > Source: an old triangle construction german book.
                  >
                  > Its suggestion to both problems is to construct b+c.
                  > I don't figure out how.
                  >
                  > I think that here is more of a matter of knowing a
                  > previous result than to put the data in a suitable position.
                  >
                  > Best regards,
                  > Luis
                  >
                  >
                  >


                  [Non-text portions of this message have been removed]
                • Luís Lopes
                  Dear Hyacinthists, Dear Jean-Pierre, Vladimir, Thank you for your solutions. I think there is a mistake in the book s suggestion. One has to calculate b-c
                  Message 8 of 9 , May 27, 2008
                    Dear Hyacinthists,

                    Dear Jean-Pierre, Vladimir,

                    Thank you for your solutions.

                    I think there is a mistake in the book's suggestion.
                    One has to calculate b-c instead.

                    Then b-c=(r_b-r_c)tan A/2

                    I didn't know this result and tried to prove it syntetically.
                    I failed and tried an algebraic approach.

                    So (B-C)/2=D and tan D=(b-c)/(r+r_a)=(r_b-r_c)/(b+c)
                    (see Court, for example).

                    One has

                    (b-c)/(r_b-r_c)=(r+r_a)/(b+c)=tan A/2

                    (b+c)/(r+r_a)=(r_b-r_c)/(b-c)=cot A/2

                    Two nice results.

                    Best regards,
                    Luis

                    ________________________________

                    To: Hyacinthos@yahoogroups.com
                    From: vndubrovsky@...
                    Date: Tue, 27 May 2008 01:59:30 +0400
                    Subject: Re: [EMHL] Triangle construction given and


                    Dear Luis

                    If you want just to construct the triangle, not necessarily using the sum
                    b+c, then it suffices to notice that
                    r_b-r_c=(b-c) cot A/2, so the problem reduces to the construction of a
                    triangle from A, r (or r_a) and b-c, which was discussed in post 16341 and
                    later.

                    Best regards,
                    Vladimir

                    2008/5/26 Luís Lopes <qed_texte@...>

                    > Dear Hyacinthists,
                    >
                    > Source: an old triangle construction german book.
                    >
                    > Its suggestion to both problems is to construct b+c.
                    > I don't figure out how.
                    >
                    > I think that here is more of a matter of knowing a
                    > previous result than to put the data in a suitable position.
                    >
                    > Best regards,
                    > Luis


                    _________________________________________________________________
                    Instale a Barra de Ferramentas com Desktop Search e ganhe EMOTICONS para o Messenger! É GRÁTIS!
                    http://www.msn.com.br/emoticonpack
                  • Vladimir Dubrovsky
                    Dear Luis, to prove b-c=(r_b-r_c)tan A/2, rewrite it as (p-c) - (p-b)=(r_b-r_c)tan A/2, where p=(a+b+c)/2. Now it suffices to prove that p-c = r_b tan A/2 and
                    Message 9 of 9 , May 27, 2008
                      Dear Luis,

                      to prove b-c=(r_b-r_c)tan A/2, rewrite it as (p-c) - (p-b)=(r_b-r_c)tan A/2,
                      where p=(a+b+c)/2. Now it suffices to prove that
                      p-c = r_b tan A/2 and a similar formula for p-b.
                      This is clear if you notice that p-c is the distance from A to the
                      contact point of the b-excircle with AC.

                      Best,
                      Vladimir
                    Your message has been successfully submitted and would be delivered to recipients shortly.