- 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.

<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] - Alexey.A.Zaslavsky wrote:
>

Dear Alexey,

> 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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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