ORTHOPOLAR CIRCLES ([EMHL] Re: ORTHOLINE)

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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 =
Message 1 of 13 , Mar 30, 2013
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.

QA-P2 = (Euler-)Poncelet Point
QA-P3 = Gergonne Steiner Point
QA-P4 = Isogonal Center
QA-P12= Orthocenter Diagonal Triangle

Best regards,

Chris van Tienhoven
www.chrisvantienhoven.nl

--- In Hyacinthos@yahoogroups.com, "Antreas" <anopolis72@...> wrote:
>
>
> 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:
>
>
> 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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