## Re: [EMHL] Steiner circumconic, X(1962)...

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• Dear Paul and Bernard ... centroids of ... anticevian ... aP is ... known, ... intersections of  ... They are the common points of the circumconic with center
Message 1 of 8 , Feb 1, 2005
Dear Paul and Bernard
> > [PY] Given a point P, can you find a point Q so that the
centroids of
> > the cevian triangle of Q is the same as the centroid of
anticevian
> > triangle of P?
> >
> > [BG]: There are at most three such points Q.
> > one of them is aP/G (cevian quotient or Ceva conjugate) where
aP is
> > the anticomplement of P.

> > *** If there are at most three such points and one of them is
known,
> > it seems that the other two can be described as the
intersections of
> > a conic and a line.

They are the common points of the circumconic with center P and the
trilinear polar of i.h.i(q) where q = aP/G, i = isotomic
conjugation, h = homothecy (G,1/4).
Friendly. Jean-Pierre
• Dear Paul and Bernard ... the ... I just realize that the line above is homothetic of the trilinear polar of q = aP/G in (G,4); this is, of course, an easier
Message 2 of 8 , Feb 2, 2005
Dear Paul and Bernard

> > > [PY] Given a point P, can you find a point Q so that the
> centroids of
> > > the cevian triangle of Q is the same as the centroid of
> anticevian
> > > triangle of P?
> > >
> > > [BG]: There are at most three such points Q.
> > > one of them is aP/G (cevian quotient or Ceva conjugate) where
> aP is
> > > the anticomplement of P.
>
> > > *** If there are at most three such points and one of them is
> known,
> > > it seems that the other two can be described as the
> intersections of
> > > a conic and a line.
>
> They are the common points of the circumconic with center P and
the
> trilinear polar of i.h.i(q) where q = aP/G, i = isotomic
> conjugation, h = homothecy (G,1/4).

I just realize that the line above is homothetic of the trilinear
polar of q = aP/G in (G,4); this is, of course, an easier
characterization.
Friendly. Jean-Pierre
• Dear Jean-Pierre and Paul, ... I have found the same thing. the 3 points also lie on three (easy to draw) conics : one of them passes through B, C, Ga (vertex
Message 3 of 8 , Feb 2, 2005
Dear Jean-Pierre and Paul,

> > > >  [PY] Given a point P, can you find a point Q so that the
> > centroids of
> > > >  the cevian triangle of Q is the same as the centroid of
> > anticevian
> > > >  triangle of P?
> > > >
> > > >  [BG]: There are at most three such points Q.
> > > >  one of them is aP/G (cevian quotient or Ceva conjugate) where
> > aP is
> > > >  the anticomplement of P.
> >
> > > >  *** If there are at most three such points and one of them is
> > known,
> > > >  it seems that the other two can be described as the
> > intersections of
> > > >  a conic and a line.
> >
> [JP] They are the common points of the circumconic with center P and
> the
> trilinear polar of i.h.i(q) where q = aP/G, i = isotomic
> conjugation, h = homothecy (G,1/4).
>
> I just realize that the line above is homothetic of the trilinear
> polar of q = aP/G in (G,4); this is, of course, an easier
> characterization.

I have found the same thing.
the 3 points also lie on three (easy to draw) conics :

one of them passes through B, C, Ga (vertex of anticomplementary
triangle) and the reflections of A in the intersections of AB, AC with
the homothetic of BC under h(P,-1/2).

Best regards

Bernard

[Non-text portions of this message have been removed]
• * was Steiner circumconic and X(1962) Dear Jean-Pierre and Bernard, [PY] Given a point P, can you find a point Q so that the centroids of the cevian triangle
Message 4 of 8 , Feb 2, 2005
* was Steiner circumconic and X(1962)

Dear Jean-Pierre and Bernard,

[PY] Given a point P, can you find a point Q so that the centroids of
the cevian triangle of Q is the same as the centroid of anticevian
triangle of P?
[BG]: There are at most three such points Q. one of them is aP/G
(cevian quotient or Ceva conjugate) where aP is the anticomplement of P.
[PY]: If there are at most three such points and one of them is
known, it seems that the other two can be described as the
intersections of a conic and a line.

[JP] They are the common points of the circumconic with center P and
the ... homothetic of the trilinear polar of q = aP/G in (G,4).

[BG]: I have found the same thing. the 3 points also lie on three
(easy to draw) conics :

one of them passes through B, C, Ga (vertex of anticomplementary
triangle) and the reflections of A in the intersections of AB, AC with
the homothetic of BC under h(P,-1/2).

*** Thank you very much for these wonderful results.

Best regards
Sincerely
Paul
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