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Re: [EMHL] Mittenpunkt And Concurrency Of Three Euler Lines

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  • Antreas P. Hatzipolakis
    ... Dear Tuan We have here two interesting locus families Let ABC be a triangle, and Q,P two points. The perpendicular to QA through P intersects AB,AC at
    Message 1 of 9 , Jun 30 11:02 PM
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      On 1-07-06, Quang Tuan Bui <bqtuan1962@...> wrote (partly):

      > Given triangle ABC, mittenpunkt Mp, incenter I. One line passing
      > through Mp perpendicular to IA cuts lines AB, AC at Ab, Ac respectively.
      > Similarly define Bc, Ba, Ca, Cb.
      > 4. The most interesting:
      > Three Euler lines of triangles MpBcCb, MpCaAc, MpAbBa are concurrent
      > at one point P.

      Dear Tuan

      We have here two interesting locus families

      Let ABC be a triangle, and Q,P two points.
      The perpendicular to QA through P intersects AB,AC at Ab,Ac, resp.
      Similarly Bc,Ba, and Ca,Cb.

      Which is the locus 1. of P 2. of Q
      such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
      are concurrent.

      Case 1 : Q = Fixed Point, P = Variable Point

      Case 2 : Q = Variable Point, P = Fixed Point

      Special Cases : P, or Q = I,O,H,G,K, etc

      And of course we can ask for concurrence of other than OH lines


      Antreas
    • Quang Tuan Bui
      Dear Antreas, Very nice idea! I try only some special case and we should do together to get some general results. I am trying with first case: Q is fixed, and
      Message 2 of 9 , Jul 1, 2006
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        Dear Antreas,
        Very nice idea! I try only some special case and we should do together to get some general results.
        I am trying with first case: Q is fixed, and line is OH. I see the locus may be nice: one line?
        The mittenpunkt may be is one very special case: Q = incenter I and the locus is only one point: mittenpunkt or some points?
        These conjectures may be the good reasons for our efforts.
        Best regards,
        Bui Quang Tuan
        PS: Please note that these only conjectures. I can confirm only with my first mittenpunkt message.

        Antreas P. Hatzipolakis <xpolakis@...> wrote:
        On 1-07-06, Quang Tuan Bui wrote (partly):

        > Given triangle ABC, mittenpunkt Mp, incenter I. One line passing
        > through Mp perpendicular to IA cuts lines AB, AC at Ab, Ac respectively.
        > Similarly define Bc, Ba, Ca, Cb.
        > 4. The most interesting:
        > Three Euler lines of triangles MpBcCb, MpCaAc, MpAbBa are concurrent
        > at one point P.

        Dear Tuan

        We have here two interesting locus families

        Let ABC be a triangle, and Q,P two points.
        The perpendicular to QA through P intersects AB,AC at Ab,Ac, resp.
        Similarly Bc,Ba, and Ca,Cb.

        Which is the locus 1. of P 2. of Q
        such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
        are concurrent.

        Case 1 : Q = Fixed Point, P = Variable Point

        Case 2 : Q = Variable Point, P = Fixed Point

        Special Cases : P, or Q = I,O,H,G,K, etc

        And of course we can ask for concurrence of other than OH lines


        Antreas



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      • Quang Tuan Bui
        Dear Antreas, I have got a very nice first result: - Q is fixed and is incenter I - Line is OH The locus (or at least all points on this line) is line
        Message 3 of 9 , Jul 1, 2006
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          Dear Antreas,
          I have got a very nice first result:
          - Q is fixed and is incenter I
          - Line is OH
          The locus (or at least all points on this line) is line connected mittenpunkt and Gergonne point.
          So mittenpunkt is NO VERY SPECIAL CASE.
          Best regards,
          Bui Quang Tuan

          Quang Tuan Bui <bqtuan1962@...> wrote:
          Dear Antreas,
          Very nice idea! I try only some special case and we should do together to get some general results.
          I am trying with first case: Q is fixed, and line is OH. I see the locus may be nice: one line?
          The mittenpunkt may be is one very special case: Q = incenter I and the locus is only one point: mittenpunkt or some points?
          These conjectures may be the good reasons for our efforts.
          Best regards,
          Bui Quang Tuan
          PS: Please note that these only conjectures. I can confirm only with my first mittenpunkt message.

          Antreas P. Hatzipolakis wrote:
          On 1-07-06, Quang Tuan Bui wrote (partly):

          > Given triangle ABC, mittenpunkt Mp, incenter I. One line passing
          > through Mp perpendicular to IA cuts lines AB, AC at Ab, Ac respectively.
          > Similarly define Bc, Ba, Ca, Cb.
          > 4. The most interesting:
          > Three Euler lines of triangles MpBcCb, MpCaAc, MpAbBa are concurrent
          > at one point P.

          Dear Tuan

          We have here two interesting locus families

          Let ABC be a triangle, and Q,P two points.
          The perpendicular to QA through P intersects AB,AC at Ab,Ac, resp.
          Similarly Bc,Ba, and Ca,Cb.

          Which is the locus 1. of P 2. of Q
          such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
          are concurrent.

          Case 1 : Q = Fixed Point, P = Variable Point

          Case 2 : Q = Variable Point, P = Fixed Point

          Special Cases : P, or Q = I,O,H,G,K, etc

          And of course we can ask for concurrence of other than OH lines


          Antreas


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        • Quang Tuan Bui
          Dear Antreas and All My Friends, Here are some results for the case: Q=incenter I and fixed, line is OH, P is any point on the line connected Gergonne point
          Message 4 of 9 , Jul 1, 2006
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            Dear Antreas and All My Friends,
            Here are some results for the case:
            Q=incenter I and fixed, line is OH, P is any point on the line connected Gergonne point and Mittenpunkt point. The concurrent point of three Euler lines as Pe.
            (These results may be helpful for general case)
            In the triangle PBcCb denote:
            circumcircle as (Oa)
            orthocenter as Ha,
            altitude foot from Bc as B'c
            altitude foot from Cb as C'b.
            Similarly with another triangles:
            PCaAc we have (Ob), Hb, C'a, A'c,
            PAbBa we have (Oc), Hc, A'b, B'a.
            Denote some intersections of lines:
            Ah = AbA'b /\ AcA'c
            Bh = BcB'c /\ BaB'a
            Ch = CaC'a /\ CbC'b
            Results:
            1. Six points: B'c, C'b, C'a, A'c, A'b, B'a are on one conic (C1)
            2. The circle (Oa) is tangent with line AbAc. Similarly for (Ob), (Oc). Other than P, these three circles cut each other at three point: Ap = (Ob) /\ (Oc), similarly define Bp, Cp. Denote circumcircle of ApBpCp as (Op).
            3. Six points: Ha, Hb, Hc, Ah, Bh, Ch are on one conic (C2).
            4. Pe is one intersection of circle (Op) and conic (C2).
            Some special nice cases:
            - When P is mittenpunkt then conic (C1) is a circle.
            - When P is centroid then conic (C2) is a circle (Op).
            Best regards,
            Bui Quang Tuan

            Quang Tuan Bui <bqtuan1962@...> wrote: Dear Antreas,
            I have got a very nice first result:
            - Q is fixed and is incenter I
            - Line is OH
            The locus (or at least all points on this line) is line connected mittenpunkt and Gergonne point.
            So mittenpunkt is NO VERY SPECIAL CASE.
            Best regards,
            Bui Quang Tuan

            Quang Tuan Bui wrote:
            Dear Antreas,
            Very nice idea! I try only some special case and we should do together to get some general results.
            I am trying with first case: Q is fixed, and line is OH. I see the locus may be nice: one line?
            The mittenpunkt may be is one very special case: Q = incenter I and the locus is only one point: mittenpunkt or some points?
            These conjectures may be the good reasons for our efforts.
            Best regards,
            Bui Quang Tuan
            PS: Please note that these only conjectures. I can confirm only with my first mittenpunkt message.

            Antreas P. Hatzipolakis wrote:
            On 1-07-06, Quang Tuan Bui wrote (partly):

            > Given triangle ABC, mittenpunkt Mp, incenter I. One line passing
            > through Mp perpendicular to IA cuts lines AB, AC at Ab, Ac respectively.
            > Similarly define Bc, Ba, Ca, Cb.
            > 4. The most interesting:
            > Three Euler lines of triangles MpBcCb, MpCaAc, MpAbBa are concurrent
            > at one point P.

            Dear Tuan

            We have here two interesting locus families

            Let ABC be a triangle, and Q,P two points.
            The perpendicular to QA through P intersects AB,AC at Ab,Ac, resp.
            Similarly Bc,Ba, and Ca,Cb.

            Which is the locus 1. of P 2. of Q
            such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
            are concurrent.

            Case 1 : Q = Fixed Point, P = Variable Point

            Case 2 : Q = Variable Point, P = Fixed Point

            Special Cases : P, or Q = I,O,H,G,K, etc

            And of course we can ask for concurrence of other than OH lines


            Antreas



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            [Non-text portions of this message have been removed]
          • Antreas P. Hatzipolakis
            Dear Tuan [BQT] ... That is, Let ABC be a triangle, and P a point. The perpendicular to IA through P intersects AB,AC at Ab,Ac, resp. Similarly Bc,Ba, and
            Message 5 of 9 , Jul 1, 2006
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              Dear Tuan

              [BQT]
              > I have got a very nice first result:
              > - Q is fixed and is incenter I
              > - Line is OH
              > The locus (or at least all points on this line) is line connected
              > mittenpunkt and Gergonne point.


              That is,

              Let ABC be a triangle, and P a point.
              The perpendicular to IA through P intersects AB,AC at Ab,Ac, resp.
              Similarly Bc,Ba, and Ca,Cb.

              The locus of P such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
              are concurrent is Mittenpunkt-Gergone Line + ???

              Probably the complete locus is some Cubic = Conic + Line


              Nice result, indeed!


              Antreas
              --
            • Antreas P. Hatzipolakis
              On 1-07-06, Quang Tuan Bui wrote: Dear Tuan ... Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line? APH
              Message 6 of 9 , Jul 1, 2006
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                On 1-07-06, Quang Tuan Bui <bqtuan1962@...> wrote:

                Dear Tuan

                > Here are some results for the case:
                > Q=incenter I and fixed, line is OH, P is any point on the line
                >connected Gergonne point and Mittenpunkt point. The concurrent point
                >of three Euler lines as Pe.

                > 4. Pe is one intersection of circle (Op) and conic (C2).

                Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line?

                APH
              • Quang Tuan Bui
                Dear Antreas, The locus of Pe as P moves on the Mit-Gerg line looks as one line, but I am not sure. I am now trying construction it. If I can construction it,
                Message 7 of 9 , Jul 1, 2006
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                  Dear Antreas,
                  The locus of Pe as P moves on the Mit-Gerg line looks as one line, but I am not sure. I am now trying construction it. If I can construction it, I can confirm.
                  Best regards,
                  Bui Quang Tuan

                  Antreas P. Hatzipolakis <xpolakis@...> wrote:
                  On 1-07-06, Quang Tuan Bui wrote:

                  Dear Tuan

                  > Here are some results for the case:
                  > Q=incenter I and fixed, line is OH, P is any point on the line
                  >connected Gergonne point and Mittenpunkt point. The concurrent point
                  >of three Euler lines as Pe.

                  > 4. Pe is one intersection of circle (Op) and conic (C2).

                  Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line?

                  APH


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                • Quang Tuan Bui
                  Dear Antreas, I can confirm: it is one line! Best regards, Bui Quang Tuan Quang Tuan Bui wrote: Dear Antreas, The locus of Pe as P moves
                  Message 8 of 9 , Jul 1, 2006
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                    Dear Antreas,
                    I can confirm: it is one line!
                    Best regards,
                    Bui Quang Tuan

                    Quang Tuan Bui <bqtuan1962@...> wrote:
                    Dear Antreas,
                    The locus of Pe as P moves on the Mit-Gerg line looks as one line, but I am not sure. I am now trying construction it. If I can construction it, I can confirm.
                    Best regards,
                    Bui Quang Tuan

                    Antreas P. Hatzipolakis wrote:
                    On 1-07-06, Quang Tuan Bui wrote:

                    Dear Tuan

                    > Here are some results for the case:
                    > Q=incenter I and fixed, line is OH, P is any point on the line
                    >connected Gergonne point and Mittenpunkt point. The concurrent point
                    >of three Euler lines as Pe.

                    > 4. Pe is one intersection of circle (Op) and conic (C2).

                    Hmmm... but what is the locus of Pe as P moves on the Mit-Gerg line?

                    APH


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                  • Antreas P. Hatzipolakis
                    Dear Tuan A variation of our locus is for Q = P That is Let ABC be a triangle, and P a point. The perpendicular to PA at P intersects AB,AC at Ab,Ac, resp.
                    Message 9 of 9 , Jul 1, 2006
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                      Dear Tuan

                      A variation of our locus is for Q = P

                      That is

                      Let ABC be a triangle, and P a point.
                      The perpendicular to PA at P intersects AB,AC at Ab,Ac, resp.
                      Similarly Bc,Ba, and Ca,Cb.

                      Which is the locus of P such that the OH lines (Euler Lines) of PBcCb, PCaAc, PAbBa
                      are concurrent?

                      APH
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