## [CEMI] Re: [EMHL] [CEMI] projective theorem

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• Dear Francois! You are right, I proved the theorem by this way. Sincerely Alexey Dear Alexey That reminds me something
Message 1 of 3 , Nov 1, 2008
Dear Francois!
You are right, I proved the theorem by this way.

Sincerely Alexey

Dear Alexey
That reminds me something already met here but I don't remember who and
when!
Maybe Jean-Pierre ?
If we choose a projective frame ABCP in which P is the unit point that is to
say an affine chart of the projective plane for which P is center of gravity
of ABC, then your theorem is equivalent to say that if Gamma is a
hyperbola through ABC and its center of gravity G, then the tangent to
Gamma at G meets both asymptots of Gamma in two points situated on the ABC
Steiner inellipse.
I note that the center O of Gamma and the symmetric T of O wrt G are also
on this ellipse.
Friendly
Francois

>
>

[Non-text portions of this message have been removed]

[Non-text portions of this message have been removed]
• Dear Alexey To sum up your sketch, we start with a triangle ABC and a point P. is the ABC -inconic with perspector P. The tripolar line L of P wrt ABC
Message 2 of 3 , Nov 1, 2008
Dear Alexey
<Gamma> is the ABC -inconic with perspector P.
The tripolar line L of P wrt ABC is also the polar line of P wrt <Gamma>.
We look at the pencil (F) of conics through ABCP and at the Desargues
involution determined by this pencil on L.
If <gamma> is a conic in (F) cutting line L in Q and R, we get a triangle
PQR autopolar wrt <Gamma>.
Then the triangle made up of the tangents at conic <gamma> in P, Q, R is
inscribed in conic <Gamma>.
Friendly
Francois

On Sat, Nov 1, 2008 at 8:07 AM, Alexey.A.Zaslavsky <zasl@...>wrote:

> Dear Francois!
> You are right, I proved the theorem by this way.
>
> Sincerely Alexey
>
>
> Dear Alexey
> That reminds me something already met here but I don't remember who and
> when!
> Maybe Jean-Pierre ?
> If we choose a projective frame ABCP in which P is the unit point that is
> to
> say an affine chart of the projective plane for which P is center of
> gravity
> of ABC, then your theorem is equivalent to say that if Gamma is a
> hyperbola through ABC and its center of gravity G, then the tangent to
> Gamma at G meets both asymptots of Gamma in two points situated on the ABC
> Steiner inellipse.
> I note that the center O of Gamma and the symmetric T of O wrt G are also
> on this ellipse.
> Friendly
> Francois
>
> >
> >
>
> [Non-text portions of this message have been removed]
>
> [Non-text portions of this message have been removed]
>
>
>

[Non-text portions of this message have been removed]
• ... Dear Alexey, Last year I have handled this subject and several other similar questions in my Gallery http://www.math.uoc.gr/~pamfilos/eGallery/Gallery.html
Message 3 of 3 , Nov 3, 2008
Alexey.A.Zaslavsky wrote:
>
> Dear Francois!
> You are right, I proved the theorem by this way.
>
> Sincerely Alexey
>
> Dear Alexey
> That reminds me something already met here but I don't remember who and
> when!
> Maybe Jean-Pierre ?
> If we choose a projective frame ABCP in which P is the unit point that
> is to
> say an affine chart of the projective plane for which P is center of
> gravity
> of ABC, then your theorem is equivalent to say that if Gamma is a
> hyperbola through ABC and its center of gravity G, then the tangent to
> Gamma at G meets both asymptots of Gamma in two points situated on the ABC
> Steiner inellipse.
> I note that the center O of Gamma and the symmetric T of O wrt G are also
> on this ellipse.
> Friendly
> Francois
>
> >
> >
>
> [Non-text portions of this message have been removed]
>
> [Non-text portions of this message have been removed]
>
>
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Dear Alexey,
Last year I have handled this subject and several other similar
questions in my Gallery
http://www.math.uoc.gr/~pamfilos/eGallery/Gallery.html
It may interest you to have a look there
I like very much projective geometry
and handle from time to time subjects
whose proofs are missing or not accessible to me.
Best regards
Paris Pamfilos
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