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Re: [EMHL] Congratulations for 10th Anniversary

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  • Ricardo Barroso
    Dear friends of Hyacinthos Congratulations for 10 years. Ricardo Barroso [Non-text portions of this message have been removed]
    Message 1 of 8 , Dec 22, 2009
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      Dear friends of Hyacinthos
      Congratulations for 10 years.

      Ricardo Barroso




      [Non-text portions of this message have been removed]
    • Antreas Hatzipolakis
      Dear Friends Thank you for your messages on Hyacinthos 10th Anniversary. Wishing you a Merry Christmas and a Happy and Heamthy New Year Antreas [Non-text
      Message 2 of 8 , Dec 22, 2009
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        Dear Friends

        Thank you for your messages on Hyacinthos 10th Anniversary.

        Wishing you a Merry Christmas and a Happy and Heamthy New Year

        Antreas


        [Non-text portions of this message have been removed]
      • fvlamoenwxs
        Dear Hyacinthians, ... I hope that everyone noted that H_A can be replaced by any point on BC. So for any point P on BC the quad formed by O, A and the
        Message 3 of 8 , Dec 28, 2009
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          Dear Hyacinthians,

          I wrote:
          > Here is a little anniversary-observation:
          > H_A be the perpendicular foot of A on BC. Let X_B and X_C be the perpendicular feet of H_A on AC and AB (yes, well known for the Taylor circle). The polygons A X_B O X_C and B X_C O X_B C have equal area (O: circumcenter).

          I hope that everyone noted that H_A can be replaced by any point on BC. So for any point P on BC the quad formed by O, A and the perpendicular feet of P on AC and AB has area half of triangle ABC.

          Kind regards,
          Floor.
        • Nikolaos Dergiades
          Dear Floor, very good problem. I had not noticed the generalization. Let the projections of P on AB, AC be D, E AF be the altitude of ABC and AF be the
          Message 4 of 8 , Dec 28, 2009
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            Dear Floor,
            very good problem.
            I had not noticed the generalization.
            Let the projections of P on AB, AC be D, E
            AF be the altitude of ABC and
            AF' be the altitude of AED.
            then the line AP is the diameter of
            AEPD and is isogonal conjugate with AF'
            and AO is isogonal conjugate with AF.
            Hence for angles
            (AO,AF') = (AP, AF) or
            (AO,ED) = (AP,BC) = f
            Hence for areas
            (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
            = R.ED.sinf = 2.(1/2).OA.ED.sinf = (AEOD)

            Best regards
            Nikos Dergiades


            [FvL]
            > I wrote:
            > > Here is a little anniversary-observation:
            > > H_A be the perpendicular foot of A on BC. Let X_B and
            > X_C be the perpendicular feet of H_A on AC and AB (yes, well
            > known for the Taylor circle). The polygons A X_B O X_C and B
            > X_C O X_B C have equal area (O: circumcenter).
            >
            > I hope that everyone noted that H_A can be replaced by any
            > point on BC. So for any point P on BC the quad formed by O,
            > A and the perpendicular feet of P on AC and AB has area half
            > of triangle ABC.
            >
            > Kind regards,
            > Floor.
            >
            >
            >
            > ------------------------------------
            >
            > Yahoo! Groups Links
            >
            >
            >     Hyacinthos-fullfeatured@yahoogroups.com
            >
            >
            >




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          • Nikolaos Dergiades
            Sorry for a typo ... the correct is Hence for areas (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf = = R.ED.sinf = 2.(1/2).OA.ED.sinf = 2.(AEOD) and (AEOD) =
            Message 5 of 8 , Dec 29, 2009
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              Sorry for a typo

              > Hence for areas
              > (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
              > = R.ED.sinf = 2.(1/2).OA.ED.sinf = (AEOD)

              the correct is

              Hence for areas
              (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
              = R.ED.sinf = 2.(1/2).OA.ED.sinf = 2.(AEOD)
              and
              (AEOD) = (1/2)ABC


              Best regards
              Nikos Dergiades





              ___________________________________________________________
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            • Francois Rideau
              And we can replace the point O by any other point O on the line AO and the (signed) area of the polygon A P_B O P_C is still constant. Happy New Year and
              Message 6 of 8 , Dec 31, 2009
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                And we can replace the point O by any other point O' on the line AO and the
                (signed) area of the polygon A P_B O' P_C is still constant.
                Happy New Year and cheer Antreas.
                Francois
                PS
                Line P_BP_C envelope a parabola tangent to the sides AB and AC and to the
                altitudes from B and C, line AO is the asymptotic direction.
                It's just a genral property of the tangents to a parabola.

                2009/12/29 Nikolaos Dergiades <ndergiades@...>

                > Sorry for a typo
                >
                > > Hence for areas
                > > (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
                > > = R.ED.sinf = 2.(1/2).OA.ED.sinf = (AEOD)
                >
                > the correct is
                >
                > Hence for areas
                > (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
                > = R.ED.sinf = 2.(1/2).OA.ED.sinf = 2.(AEOD)
                > and
                > (AEOD) = (1/2)ABC
                >
                >
                > Best regards
                > Nikos Dergiades
                >
                >
                >
                >
                >
                > ___________________________________________________________
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                > ���������� �� ���������� �������� (spam); �� Yahoo! Mail
                > �������� ��� �������� ������ ��������� ���� ��� �����������
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                >
                >
                >
                > ------------------------------------
                >
                > Yahoo! Groups Links
                >
                >
                >
                >


                [Non-text portions of this message have been removed]
              • Francois Rideau
                I forget to say that H_A is of course the focus of the parabola. Francois 2009/12/31 Francois Rideau ... [Non-text portions of
                Message 7 of 8 , Dec 31, 2009
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                  I forget to say that H_A is of course the focus of the parabola.
                  Francois

                  2009/12/31 Francois Rideau <francois.rideau@...>

                  > And we can replace the point O by any other point O' on the line AO and the
                  > (signed) area of the polygon A P_B O' P_C is still constant.
                  > Happy New Year and cheer Antreas.
                  > Francois
                  > PS
                  > Line P_BP_C envelope a parabola tangent to the sides AB and AC and to the
                  > altitudes from B and C, line AO is the asymptotic direction.
                  > It's just a genral property of the tangents to a parabola.
                  >
                  > 2009/12/29 Nikolaos Dergiades <ndergiades@...>
                  >
                  > Sorry for a typo
                  >>
                  >> > Hence for areas
                  >> > (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
                  >> > = R.ED.sinf = 2.(1/2).OA.ED.sinf = (AEOD)
                  >>
                  >> the correct is
                  >>
                  >> Hence for areas
                  >> (ABC) = (1/2).BC.AP.sinf = R.sinA.AP.sinf =
                  >> = R.ED.sinf = 2.(1/2).OA.ED.sinf = 2.(AEOD)
                  >> and
                  >> (AEOD) = (1/2)ABC
                  >>
                  >>
                  >> Best regards
                  >> Nikos Dergiades
                  >>
                  >>
                  >>
                  >>
                  >>
                  >> ___________________________________________________________
                  >> �������������� Yahoo!;
                  >> ���������� �� ���������� �������� (spam); �� Yahoo! Mail
                  >> �������� ��� �������� ������ ��������� ���� ��� �����������
                  >> ��������� http://login.yahoo.com/config/mail?.intl=gr
                  >>
                  >>
                  >>
                  >> ------------------------------------
                  >>
                  >> Yahoo! Groups Links
                  >>
                  >>
                  >>
                  >>
                  >


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