- Dear friends,

it is known the Fregier's theorem:

If AXY is a right agnled triangle at A

and is inscribed in a conic Co with A constant and XY variable then

XY passes through a constant point P.

We call this point P the Fregier point Fr(A) of A relative

to this conic Co.

If A moves on Co then its Fregier point moves on another conic

that we call the Fregier conic of Co.

Now if Co is a circumconic of triangle ABC with equation in barycentrics

pyz + qzx + rxy = 0 then the Fregier conic of this circumconic is a circle

if the point (p : q : r) lies on the orthic axis of ABC.

If the circumconic Co is the Steiner circumellipse and

P = (x : y : z) in bars

is a point on it then the Fregier point of P is

Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)

and the Fregier conic of the Co is an ellipse homothetic to Co

with center G and ratio of homothecy

2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).

I think that from the 23 known in ETC points of the

Steiner circumellipse only two have Fregier points on ETC that is:

Fr( X(99) ) = X(69)

Fr( X(671) ) = X(1992).

Best regards

Nikos Dergiades - Of course, you must read:

Let S be a quadric in a n-dimensional euclidian space and M be some fixed

point on S

Let (Mx_1, Mx_2, ..., Mx_n) n lines through M mutually orthogonal.

They meet again S in n other points A_1, A_2,..., A_n.

Then the hyperplane on A_1, A_2, ..., A_n is on some fixed point H

belonging to the normal at S on *M*.

The locus of the Fregier point H is another quadric similar to S.

Friendly

On Sat, Sep 8, 2012 at 12:38 PM, Francois Rideau

<francois.rideau@...>wrote:

> Another remark:

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

> The Fregier theorem is still true in any dimension under some general

> conditions.

> Let S be a quadric in a n-dimensional euclidian space and M be some fixed

> point on S

> Let (Mx_1, Mx_2, ..., Mx_n) n lines through M mutually orthogonal.

> They meet again S in n other points A_1, A_2,..., A_n.

> Then the hyperplane on A_1, A_2, ..., A_n is on some fixed point H

> belonging to the normal at S on h.

> The locus of the Fregier point H is another quadric similar to S.

> Friendly

> Francois

> PS

> Of course, some calculations is needed for its proof!

>

>

>

> On Sat, Sep 8, 2012 at 12:27 PM, Francois Rideau <

> francois.rideau@...> wrote:

>

>> Dear Nikos

>> The best to get the Fregier conic is to make calculations on a reduced

>> form, for example: x²/a²+y²/b²-1 = 0 for an ellipse or y² - 2px = 0 for a

>> parabola or xy = k for a hyperbola.

>> If you want a more synthetic way, look at what happens when the sides of

>> the right angle through point M on the conic are parallel to the axis of

>> this conic.

>> Friendly

>> Francois

>>

>>

>> On Fri, Sep 7, 2012 at 11:40 AM, Nikolaos Dergiades <ndergiades@...>wrote:

>>

>>> Dear Francois,

>>> I wrote that if the perspector of a circumconic

>>> lies on the Orthic axis then its Fregier conic

>>> satisfies the condition to be a circle.

>>> I did not observed that there is more.

>>> The Fregier conic satisfies and other conditions

>>> and I think that is the line at infinity.

>>>

>>> If my sketch is correct then if the perspector

>>> of a circumconic lies on the orthic axis or

>>> if the center of the circumconic lies on the NPC

>>> then the conic is a rectangular hyperbola Co

>>> and the hypotenuse XY of the right angled triangle

>>> AXY inscribed in Co is parallel to the normal

>>> of Co at A. Hence the Fregier conic of Co is the line

>>> at infinity.

>>> Best regards

>>> Nikos Dergiades

>>>

>>>

>>>

>>>

>>> > Dear Nikos

>>> > I have always thought that the Fregier conic of a conic

>>> > Gamma was

>>> > (indirectly) similar to Gamma.

>>> > Moreover circumconics with perspectors on orthic axis are

>>> > rectangular

>>> > hyperbola and so their Fregier conics are also

>>> > rectangular hyperbola.

>>> > Friendly Francois

>>> >

>>> > On Thu, Sep 6, 2012 at 8:24 PM, Nikolaos Dergiades <

>>> ndergiades@...>wrote:

>>> >

>>> > > **

>>> > >

>>> > >

>>> > > Dear friends,

>>> > > it is known the Fregier's theorem:

>>> > > If AXY is a right agnled triangle at A

>>> > > and is inscribed in a conic Co with A constant and XY

>>> > variable then

>>> > > XY passes through a constant point P.

>>> > >

>>> > > We call this point P the Fregier point Fr(A) of A

>>> > relative

>>> > > to this conic Co.

>>> > > If A moves on Co then its Fregier point moves on

>>> > another conic

>>> > > that we call the Fregier conic of Co.

>>> > >

>>> > > Now if Co is a circumconic of triangle ABC with

>>> > equation in barycentrics

>>> > > pyz + qzx + rxy = 0 then the Fregier conic of this

>>> > circumconic is a circle

>>> > > if the point (p : q : r) lies on the orthic axis of

>>> > ABC.

>>> > >

>>> > > If the circumconic Co is the Steiner circumellipse and

>>> > > P = (x : y : z) in bars

>>> > > is a point on it then the Fregier point of P is

>>> > > Fr(P)=(xSA+ySC+zSB : xSC+ySB+zSA : xSB+ySA+zSC)

>>> > > and the Fregier conic of the Co is an ellipse

>>> > homothetic to Co

>>> > > with center G and ratio of homothecy

>>> > > 2.sqrt[a^4+b^4+c^4-(bc)^2-(ca)^2-(ab)^2] / (aa+bb+cc).

>>> > > I think that from the 23 known in ETC points of the

>>> > > Steiner circumellipse only two have Fregier points on

>>> > ETC that is:

>>> > > Fr( X(99) ) = X(69)

>>> > > Fr( X(671) ) = X(1992).

>>> > >

>>> > > Best regards

>>> > > Nikos Dergiades

>>> > >

>>> > >

>>> > >

>>> >

>>> >

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

>>> >

>>> >

>>> >

>>> > ------------------------------------

>>> >

>>> > Yahoo! Groups Links

>>> >

>>> >

>>> > Hyacinthos-fullfeatured@yahoogroups.com

>>> >

>>> >

>>>

>>>

>>> ------------------------------------

>>>

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>>>

>>>

>>>

>>>

>>

>