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ORTHOPOLAR CIRCLES ([EMHL] Re: ORTHOLINE)

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  • 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 1 of 13 , Mar 30, 2013
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      --- 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 2 of 13 , Mar 30, 2013
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        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
        >
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