## Re: [EMHL] Two Questions

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• Dear Eisso, ... this is the Orion transform in Clark s ETC. See also the thread : Another stellar (or flowered) transformation, Hyacinthos #7999 & sq. The
Message 1 of 4 , Mar 17, 2008
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Dear Eisso,

> [EJA] as I was playing with some Cevian triangles I noticed the
> following:
>
> I) Let A'B'C' be the Cevian triangle of a point P w.r.t. a triangle
> ABC. Now let P[A] be the reflection of P in B'C' and P[B] and P[C]
> defined analogously. Then AP[A], BP[B], and CP[C] are concurrent in a
> point P', thus defining a (birational) transformation P to P'.
>
> II) The pedal points of the altitudes of A'B'C' each lie on the
> corresponding cevian XP[X], i.e. the pedal point of the altitude
> from A'
> on B'C' lies on AP[A] and so on.
>
> With regard to I), I would like to know if this transformation has
> been
> studied at all and perhaps even has a name. Regarding II), I am just
> wondering if this property is known.

this is the Orion transform in Clark's ETC.

Another stellar (or flowered) transformation, Hyacinthos #7999 & sq.

The fixed points of this transformation are the CPCC points in

http://pagesperso-orange.fr/bernard.gibert/Tables/table11.html

Best regards

Bernard

[Non-text portions of this message have been removed]
• Dear Eisso ... triangle ... in a ... from A ... been ... just ... Look at #8039; you ll see a proof of your results (your point P is the isogonal conjugate
Message 2 of 4 , Mar 17, 2008
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Dear Eisso
> as I was playing with some Cevian triangles I noticed the following:
>
> I) Let A'B'C' be the Cevian triangle of a point P w.r.t. a
triangle
> ABC. Now let P[A] be the reflection of P in B'C' and P[B] and P[C]
> defined analogously. Then AP[A], BP[B], and CP[C] are concurrent
in a
> point P', thus defining a (birational) transformation P to P'.
>
> II) The pedal points of the altitudes of A'B'C' each lie on the
> corresponding cevian XP[X], i.e. the pedal point of the altitude
from A'
> on B'C' lies on AP[A] and so on.
>
> With regard to I), I would like to know if this transformation has
been
> studied at all and perhaps even has a name. Regarding II), I am
just
> wondering if this property is known.

Look at #8039; you'll see a proof of your results (your point P' is
the isogonal conjugate of P wrt the orthic triangle of A'B'C')
Clark Kimberling named P->P' the Orion transformation and the first
result above was named the Begonia theorem by Darij Grinberg who gave
a proof available at
http://de.geocities.com/darij_grinberg/Begonia.zip
The transformation is rational but not birational (in fact there are
7 points P -not necessarily real- with a given image P')
Friendly. Jean-Pierre
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