## Mantel * Noyer * Droz-Farny * Goormaghtigh

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• Dear Hyacinthists, an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh  or the Simson-Wallace generalized’’ has been put on my website.
Message 1 of 5 , Sep 3, 2012
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Dear Hyacinthists,
an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh  or the Simson-Wallace generalized’’ has been put on my website.

http://perso.orange.fr/jl.ayme%c2%a0%c2%a0%c2%a0 vol. 12

Sincerely
Jean-Louis

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• Dear Jean-Louis! an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh or the Simson-Wallace generalized’’ has been put on my website.
Message 2 of 5 , Sep 3, 2012
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Dear Jean-Louis!

an article concerning “Mantel * Noyer * Droz-Farny * Goormaghtigh or the Simson-Wallace generalized’’ has been put on my website.

http://perso.orange.fr/jl.ayme vol. 12

Some years ago C.Pohoata and N.Beluhov independntly found a new proof of Goormaghtigh theorem: the line A'B' touches the inconic with center P. Partially if P is the incenter then This line touches the incircle.

Sincerely Alexey

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• Hello Is this result known? Given a quadrilateral ABCD, and the parabola tangent to its sides, any other tangent to the parabola divides the oposit sides of
Message 3 of 5 , Sep 4, 2012
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Hello

Is this result known?

Given a quadrilateral ABCD, and the parabola tangent to its sides, any
other tangent to the parabola divides the oposit sides of ABCD in the
same ratio.

Thanks

Martin
• Dear Martin Yes; it is known for a long long time. Given a conic Gamma and two of its tangent L and L ; a variable tangent T meets L in m and L in m . Then it
Message 4 of 5 , Sep 5, 2012
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Dear Martin
Yes; it is known for a long long time.
Given a conic Gamma and two of its tangent L and L'; a variable tangent T
meets L in m and L' in m'.
Then it is known for long that the correspondance between m and m' is
projective or homographic.
In case Gamma is a parabola, the line of infinity is a tangent; hence
points at infinity of line L and L' are in correspondence or homologuous,
that is to say the correspondence between m and m' is affine and so
preserves ratio.
Friendly
Francois

On Tue, Sep 4, 2012 at 2:21 PM, Mart�n Acosta <maedu@...> wrote:

> **
>
>
> Hello
>
> Is this result known?
>
> Given a quadrilateral ABCD, and the parabola tangent to its sides, any
> other tangent to the parabola divides the oposit sides of ABCD in the
> same ratio.
>
> Thanks
>
> Martin
>
>

[Non-text portions of this message have been removed]
• This also follows from some results about cross-ratio. Cross-ratio is defined originally for 4 collinear points, and then for 4 concurrent lines, then for 4
Message 5 of 5 , Sep 6, 2012
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This also follows from some results about cross-ratio. Cross-ratio is defined
originally for 4 collinear points, and then for 4 concurrent lines, then for 4 points on
a conic, and then for 4 tangents to a conic. This last case gives the following lemma:

Let lines a,c,e,f,x be 5 tangents to a conic. The cross-ratio R(a,c;e,f) is
(a.x - e.x)*(c.x - f.x)/((a.x - f.x)*(c.x - e.x))
where p.q means the point of intersection of lines p and q, and P - Q means
the directed distance from point Q to point P. The lemma is that changing x
to any tangent of this conic does not change the value of R.

If the conic is a parabola, then the line at infinity is a tangent line.
Put f = the line at infinity, so R simplifies to R' = (a.x - e.x)/(c.x - e.x).
Then -R' is the ratio in which e.x divides the segment of x between a.x and c.x

Let your quadrilateral have sides a,b,c,d in cyclic order, and let e be the other tangent
line. Then x=b and x=d both give the same value of -R', which is just your result.
--
Barry Wolk

> From: Francois Rideau <francois.rideau@...>
>
> Dear Martin
> Yes; it is known for a long long time.
> Given a conic Gamma and two of its tangent L and L'; a variable tangent T
> meets L in m and L' in m'.
> Then it is known for long that the correspondance between m and m' is
> projective or homographic.
> In case Gamma is a parabola, the line of infinity is a tangent; hence
> points at infinity of line L and L' are in correspondence or homologuous,
> that is to say the correspondence between m and m' is affine and so
> preserves ratio.
> Friendly
> Francois
>
>>
>> Hello
>>
>> Is this result known?
>>
>> Given a quadrilateral ABCD, and the parabola tangent to its sides, any
>> other tangent to the parabola divides the oposit sides of ABCD in the
>> same ratio.
>>
>> Thanks
>>
>> Martin
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