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

ORTHOPOLAR CIRCLES ([EMHL] Re: ORTHOLINE)

Expand Messages
  • Antreas
    Let ABC be a triangle, P, Q two points and O,Q1,Q2,Q3 the circumcenters of ABC, QBC, QCA, QAB, resp. Denote: P0 = the orthopole of PO wrt ABC P1 = the
    Message 1 of 13 , Mar 30, 2013
    • 0 Attachment
      Let ABC be a triangle, P, Q two points and O,Q1,Q2,Q3 the circumcenters of ABC,
      QBC, QCA, QAB, resp.

      Denote:

      P0 = the orthopole of PO wrt ABC

      P1 = the orthopole of PQ1 wrt QBC

      P2 = the orthopole of PQ2 wrt QCA

      P3 = the orthopole of PQ3 wrt QAB

      We have:

      1. P0, P1, P2, P3 lie on the NPCs (N),(N1),(N2),(N3) of ABC,
      QBC, QCA, QAB, resp. (since the respective lines pass through the circumcenters
      of the respective triangles)

      2. The NPCs of ABC, QBC, QCA, QAB concur at the Poncelet point Q*
      of Q wrt ABC.

      CONJECTURE:

      The points P0, P1, P2, P3, Q* are concyclic.

      Figure:

      http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles_30.html


      Antreas


      --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
      >
      > [APH]
      > > Conjecture:
      > > Let P be a point on the Euler line of ABC, and O, Oa,Ob,Oc
      > > the circumcenters of ABC, IBC, ICA, IAB.
      > > The orthopoles of PO [=Euler line of ABC], POa, POb, POc wrt
      > > ABC, IBC, ICA, IAB, resp.
      > > are concyclic.
      > >
      > > Question: Is it true for any point Q (instead of I) on the
      > > Neuberg cubic?
      >
      > If the Conjecture is true (as I think), then we have an interesting
      > locus: As P moves on the Euler Line of ABC, all circles pass through
      > the orthopole of the Euler Line wrt ABC.
      >
      > Which is the locus of their centers?
      >
      > http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles.html
      >
      > APH
      >
    • Antreas
      For quadrilateral: Let ABCD be a quadrilateral, A ,B ,C ,D the circumcenters of BCD, CDA, DAB, ABC, resp. and P a point. Denote: 1 = the orthopole of PA wrt
      Message 2 of 13 , Mar 30, 2013
      • 0 Attachment
        For quadrilateral:

        Let ABCD be a quadrilateral, A',B',C',D' the circumcenters of BCD,
        CDA, DAB, ABC, resp. and P a point.

        Denote:

        1 = the orthopole of PA' wrt BCD

        2 = the orthopole of PB' wrt CDA

        3 = the orthopole of PC' wrt DAB

        4 = the orthopole of PD' wrt ABC

        Conjecture:

        The points 1,2,3,4 are concyclic. The circle passes through the Poncelet point of ABCD (=the point where the NPCs of BCD, CDA, DAB,
        ABC concur)

        Figure:

        http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles-quadrilateral.html

        Antreas

        [APH]
        > Let ABC be a triangle, P, Q two points and O,Q1,Q2,Q3 the circumcenters of ABC,
        > QBC, QCA, QAB, resp.
        >
        > Denote:
        >
        > P0 = the orthopole of PO wrt ABC
        >
        > P1 = the orthopole of PQ1 wrt QBC
        >
        > P2 = the orthopole of PQ2 wrt QCA
        >
        > P3 = the orthopole of PQ3 wrt QAB
        >
        > We have:
        >
        > 1. P0, P1, P2, P3 lie on the NPCs (N),(N1),(N2),(N3) of ABC,
        > QBC, QCA, QAB, resp. (since the respective lines pass through the circumcenters
        > of the respective triangles)
        >
        > 2. The NPCs of ABC, QBC, QCA, QAB concur at the Poncelet point Q*
        > of Q wrt ABC.
        >
        > CONJECTURE:
        >
        > The points P0, P1, P2, P3, Q* are concyclic.
        >
        > Figure:
        >
        > http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles_30.html
        >
        >
        > Antreas
      • Antreas
        ... If ABCD is cyclic (PO = PA = PB = PC := L , a line passing through the circumcenter O of ABCD), the circle (1,2,3,4) is a line. APH
        Message 3 of 13 , Mar 30, 2013
        • 0 Attachment
          --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
          >
          > For quadrilateral:
          >
          > Let ABCD be a quadrilateral, A',B',C',D' the circumcenters of BCD,
          > CDA, DAB, ABC, resp. and P a point.
          >
          > Denote:
          >
          > 1 = the orthopole of PA' wrt BCD
          >
          > 2 = the orthopole of PB' wrt CDA
          >
          > 3 = the orthopole of PC' wrt DAB
          >
          > 4 = the orthopole of PD' wrt ABC
          >
          > Conjecture:
          >
          > The points 1,2,3,4 are concyclic. The circle passes through the Poncelet point of ABCD (=the point where the NPCs of BCD, CDA, DAB,
          > ABC concur)
          >
          > Figure:
          >
          > http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles-quadrilateral.html
          >
          > Antreas

          If ABCD is cyclic (PO = PA' = PB' = PC' := L , a line passing through
          the circumcenter O of ABCD), the circle (1,2,3,4) is a line.

          APH
        • Chris Van Tienhoven
          Dear Antreas, Very nice observation! Let s call the circle through points 1,2,3,4 the QA-Orthopole-circle. Then: when P = QA-P3 then the QA-Orthopole-circle =
          Message 4 of 13 , Mar 30, 2013
          • 0 Attachment
            Dear Antreas,

            Very nice observation!
            Let's call the circle through points 1,2,3,4 the QA-Orthopole-circle.
            Then:
            when P = QA-P3 then the QA-Orthopole-circle = circle with diameter QA-P2.QA-P3,
            when P = QA-P4 then the QA-Orthopole-circle = point QA-P2,
            when P = QA-P12 then the QA-Orthopole-circle = circumcircle of the Diagonal Triangle.

            Explanation of above Quadrangle Points:
            QA-P2 = (Euler-)Poncelet Point
            QA-P3 = Gergonne Steiner Point
            QA-P4 = Isogonal Center
            QA-P12= Orthocenter Diagonal Triangle
            More information at http://www.chrisvantienhoven.nl/quadrangle-objects/17-mathematics/4-quadrangle-objects.html

            Best regards,

            Chris van Tienhoven
            www.chrisvantienhoven.nl


            --- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
            >
            > For quadrilateral:
            >
            > Let ABCD be a quadrilateral, A',B',C',D' the circumcenters of BCD,
            > CDA, DAB, ABC, resp. and P a point.
            >
            > Denote:
            >
            > 1 = the orthopole of PA' wrt BCD
            >
            > 2 = the orthopole of PB' wrt CDA
            >
            > 3 = the orthopole of PC' wrt DAB
            >
            > 4 = the orthopole of PD' wrt ABC
            >
            > Conjecture:
            >
            > The points 1,2,3,4 are concyclic. The circle passes through the Poncelet point of ABCD (=the point where the NPCs of BCD, CDA, DAB,
            > ABC concur)
            >
            > Figure:
            >
            > http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles-quadrilateral.html
            >
            > Antreas
            >
            > [APH]
            > > Let ABC be a triangle, P, Q two points and O,Q1,Q2,Q3 the circumcenters of ABC,
            > > QBC, QCA, QAB, resp.
            > >
            > > Denote:
            > >
            > > P0 = the orthopole of PO wrt ABC
            > >
            > > P1 = the orthopole of PQ1 wrt QBC
            > >
            > > P2 = the orthopole of PQ2 wrt QCA
            > >
            > > P3 = the orthopole of PQ3 wrt QAB
            > >
            > > We have:
            > >
            > > 1. P0, P1, P2, P3 lie on the NPCs (N),(N1),(N2),(N3) of ABC,
            > > QBC, QCA, QAB, resp. (since the respective lines pass through the circumcenters
            > > of the respective triangles)
            > >
            > > 2. The NPCs of ABC, QBC, QCA, QAB concur at the Poncelet point Q*
            > > of Q wrt ABC.
            > >
            > > CONJECTURE:
            > >
            > > The points P0, P1, P2, P3, Q* are concyclic.
            > >
            > > Figure:
            > >
            > > http://anthrakitis.blogspot.gr/2013/03/orthopolar-circles_30.html
            > >
            > >
            > > Antreas
            >
          Your message has been successfully submitted and would be delivered to recipients shortly.