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antiorthic axis of excentral triangle

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  • rhutson2
    Dear Hyacinthists: Does anyone have some magic to find any ETC centers that lie on the antiorthic axis of the excentral triangle? The best I can find is that
    Message 1 of 6 , May 25, 2012
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      Dear Hyacinthists:

      Does anyone have some magic to find any ETC centers that lie on the antiorthic axis of the excentral triangle? The best I can find is that it meets the line at infinity at the isogonal conjugate of X(3659).

      The crossdifference of every pair of points on this line (i.e., the isogonal conjugate of the trilinear pole of this line) has the nice trilinears:

      csc B/2 + csc C/2 : :

      Thanks in advance,
      Randy
    • Chris Van Tienhoven
      Dear Randy, I can t find any in the range X(1) - X(3514). Best regards, Chris van Tienhoven
      Message 2 of 6 , May 25, 2012
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        Dear Randy,

        I can't find any in the range X(1) - X(3514).
        Best regards,

        Chris van Tienhoven


        --- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
        >
        > Dear Hyacinthists:
        >
        > Does anyone have some magic to find any ETC centers that lie on the antiorthic axis of the excentral triangle? The best I can find is that it meets the line at infinity at the isogonal conjugate of X(3659).
        >
        > The crossdifference of every pair of points on this line (i.e., the isogonal conjugate of the trilinear pole of this line) has the nice trilinears:
        >
        > csc B/2 + csc C/2 : :
        >
        > Thanks in advance,
        > Randy
        >
      • Randy Hutson
        Thanks Chris. What about the de Longchamps line of the excentral triangle (line at infinity intercept = X(513), or the Gergonne line of the excentral triangle?
        Message 3 of 6 , May 28, 2012
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          Thanks Chris.

          What about the de Longchamps line of the excentral triangle (line at infinity intercept = X(513), or the Gergonne line of the excentral triangle?

          Best regards,
          Randy




          >________________________________
          > From: Chris Van Tienhoven <van10hoven@...>
          >To: Hyacinthos@yahoogroups.com
          >Sent: Friday, May 25, 2012 4:59 PM
          >Subject: [EMHL] Re: antiorthic axis of excentral triangle
          >
          >

          >Dear Randy,
          >
          >I can't find any in the range X(1) - X(3514).
          >Best regards,
          >
          >Chris van Tienhoven
          >
          >--- In Hyacinthos@yahoogroups.com, "rhutson2" <rhutson2@...> wrote:
          >>
          >> Dear Hyacinthists:
          >>
          >> Does anyone have some magic to find any ETC centers that lie on the antiorthic axis of the excentral triangle? The best I can find is that it meets the line at infinity at the isogonal conjugate of X(3659).
          >>
          >> The crossdifference of every pair of points on this line (i.e., the isogonal conjugate of the trilinear pole of this line) has the nice trilinears:
          >>
          >> csc B/2 + csc C/2 : :
          >>
          >> Thanks in advance,
          >> Randy
          >>
          >
          >
          >
          >
          >

          [Non-text portions of this message have been removed]
        • Chris Van Tienhoven
          Dear Randy, [RH] ... I found that the de Longchamps line of the excentral triangle passes through (points in the range X(1)-X(3514): X(239) X(514) X(649)
          Message 4 of 6 , May 28, 2012
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            Dear Randy,


            [RH]
            >
            > What about the de Longchamps line of the excentral triangle (line at infinity intercept = X(513), or the Gergonne line of the excentral triangle?

            I found that the de Longchamps line of the excentral triangle passes through (points in the range X(1)-X(3514):
            X(239)
            X(514)
            X(649)
            X(1019)
            X(1021)
            X(3218)
            I also found a point outside the range X(1)-X(3514):
            X(4560)
            That's because it was in my personal database before it was registered in ETC.
            Special is too that this line is the perpendicular bisector of:
            X(1276), X(1277).
            The coefficients of this line are: (b+c : c+a : a+b).
            Hmmm, that's pretty spectacular ....... such simple coefficients....
            I could not find it in the list of central lines.


            I could not find any ETC-points in the range X(1)-X(3514) on the Gergonne Line of the Excentral Triangle.
            However these intersection points should lie on this line:
            X40.X164 ^ X348.X1216 (= X241 Excentral Triangle)
            X44.X513 ^ X89.X3423 (= X650 Excentral Triangle)
            X3.X266 ^ X9.X363 ^ X165.X503 (= X1465 Excentral Triangle)
            X84.X2484 ^ X508.X2036 ^ X566.X1433 (= X1638 Excentral Triangle)
            I hope these intersection points were secure measurements.


            Best regards,

            Chris van Tienhoven
          • Randy Hutson
            Thanks Chris, but I think these points (line X(239)X(514)) lie on the Lemoine axis of the excentral triangle, not the de Longchamps line of e.t. Randy ...
            Message 5 of 6 , May 28, 2012
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              Thanks Chris, but I think these points (line X(239)X(514)) lie on the Lemoine axis of the excentral triangle, not the de Longchamps line of e.t.


              Randy




              >________________________________
              > From: Chris Van Tienhoven <van10hoven@...>
              >To: Hyacinthos@yahoogroups.com
              >Sent: Monday, May 28, 2012 12:39 PM
              >Subject: [EMHL] Re: antiorthic axis of excentral triangle
              >
              >

              >Dear Randy,
              >
              >[RH]
              >>
              >> What about the de Longchamps line of the excentral triangle (line at infinity intercept = X(513), or the Gergonne line of the excentral triangle?
              >
              >I found that the de Longchamps line of the excentral triangle passes through (points in the range X(1)-X(3514):
              >X(239)
              >X(514)
              >X(649)
              >X(1019)
              >X(1021)
              >X(3218)
              >I also found a point outside the range X(1)-X(3514):
              >X(4560)
              >That's because it was in my personal database before it was registered in ETC.
              >Special is too that this line is the perpendicular bisector of:
              >X(1276), X(1277).
              >The coefficients of this line are: (b+c : c+a : a+b).
              >Hmmm, that's pretty spectacular ....... such simple coefficients....
              >I could not find it in the list of central lines.
              >
              >I could not find any ETC-points in the range X(1)-X(3514) on the Gergonne Line of the Excentral Triangle.
              >However these intersection points should lie on this line:
              >X40.X164 ^ X348.X1216 (= X241 Excentral Triangle)
              >X44.X513 ^ X89.X3423 (= X650 Excentral Triangle)
              >X3.X266 ^ X9.X363 ^ X165.X503 (= X1465 Excentral Triangle)
              >X84.X2484 ^ X508.X2036 ^ X566.X1433 (= X1638 Excentral Triangle)
              >I hope these intersection points were secure measurements.
              >
              >Best regards,
              >
              >Chris van Tienhoven
              >
              >
              >
              >
              >

              [Non-text portions of this message have been removed]
            • Chris Van Tienhoven
              Dear Randy, ... You are quite right. Sorry for the mistake! For the De lonchamps Line of the Ecentral Triangle I find (just like you) only X(513) in the range
              Message 6 of 6 , May 28, 2012
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                Dear Randy,

                --- In Hyacinthos@yahoogroups.com, Randy Hutson <rhutson2@...> wrote:
                >
                > I think these points (line X(239)X(514)) lie on the Lemoine axis of the excentral triangle, not the de Longchamps line of e.t.

                You are quite right. Sorry for the mistake!
                For the De lonchamps Line of the Ecentral Triangle I find (just like you) only X(513) in the range X(1)-X(3514).

                Best regards,

                Chris
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