## [CEMI] Re: [EMHL] [CEMI] octagon

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• Dear Francois! Thank you for interesting idea. I proved this theorem using the projective map transforming the line pqr to the infinite line. Also it can be
Message 1 of 1 , Sep 30, 2008
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Dear Francois!
Thank you for interesting idea. I proved this theorem using the projective map transforming the line pqr to the infinite line. Also it can be obtained from some general facts of agebraic geometry.

Sincerely Alexey

Dear Alexey
I did not know this theorem.
You take any 8 points on a same conic and you note your intersections:
p = (12,45), q= (34,67), r = (23,78), s= (56,81)
Now you move point 8 on the conic and look at what happens.
As points r and s are on some homographic range on the lines 23 and 56, line
rs envelopes some conic tangent to the lines 12, 13, 23; 56, 57, 67 and pq,
that's what you are meaning?
Is there some duality around here, for starting from 8 points on a same
conic, we get 8 lines tangent to another one!
Friendly
Francois

On Fri, Sep 26, 2008 at 12:55 PM, Alexey.A.Zaslavsky <zasl@...>wrote:

> Dear colleagues!
> Is next fact known?
> Let 8 points lie on the conic. Consider the common points of lines 12 and
> 45, 34 and 67, 56 and 81, 78 and 23. If three of these four points are
> collinear then the fourth lies on the same line. In this case four common
> points of 12 and 67 etc. also are collinear.
>
> Sincerely Alexey
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>
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>
>
>

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