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ORTHOPOLAR CIRCLES ([EMHL] Re: ORTHOLINE)

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  • 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 1 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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