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• Foci inner Steiner ellipse in circular geometry

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• Dear friends At the moment, I look at applications of circular geometry in my favourite choo-choo theory. In triangle geometry, I only know isodynamic points
Message 1 of 10 , Nov 11, 2007
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Dear friends

At the moment, I look at applications of circular geometry in my favourite
choo-choo theory.
In triangle geometry, I only know isodynamic points as centers defined by
circular geometry.
As a result of a general theorem I found in choo-choo theory, I give you a
circular mean to determine foci of the Steiner inner ellipse.
So we start with our usual beloved triangle ABC, its circumcenter O and its
Lemoine point K.
By the way, why this damned Lemoine point is always labelled K ?
Let A', B', C' respectively the orthogonal projections of O onto the
symedians AK, BK, CK. ( They are on the Brocard circle.)
Let f be the direct circular map such that: f(A) = A', f(B) = B', f(C) =
C'.
Then f is of order 2, that is to say involutive. (Do you see the
relationship with one of my previous post?)
Its central point is the centroid G: f(G) = Infty and f(Infty) = G.
Its fixed points F and F' are the foci of the inner Steiner ellipse.
In other words, we have harmonic quadrangles: (A, A', F, F'), (B, B', F,
F'), (C, C', F, F'), so giving a simple construction of these foci.
Besides as nothing is new in triangle geometry, I am sure these quadrangles
are known for a long time.

I am eager to know if similar results, using circular geometry, are known
in triangle geometry.

In a few days , I will give you a GSP-Cabri tool giving the fixed points of
a direct circular map, so you can check this property.

Friendly
Francois

[Non-text portions of this message have been removed]
• Dear Francois , ... L is the de Longchamps point... ... A B C is the second Brocard triangle. ... I suspect that your map is the one described in
Message 1 of 10 , Nov 11, 2007
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Dear Francois ,

> So we start with our usual beloved triangle ABC, its circumcenter O
> and its
> Lemoine point K.
> By the way, why this damned Lemoine point is always labelled K ?

L is the de Longchamps point...

> Let A', B', C' respectively the orthogonal projections of O onto the
> symedians AK, BK, CK. ( They are on the Brocard circle.)

A'B'C' is the second Brocard triangle.

> Let f be the direct circular map such that: f(A) = A', f(B) = B', f
> (C) =
> C'.
> Then f is of order 2, that is to say involutive.

I suspect that your map is the one described in "orthocorrespondence
and orthopivotal cubics" §5.

f(M) is the product of an inversion and a reflection. It is also :

- the center of the polar conic of M wrt the McCay or Kjp cubic,

- the isoconjugate of M in the triangle formed by G and the circular
points at infinity. In other words, the pole of M in the pencil of
rectangular hyperbolas passing through the 4 foci of the Steiner
inellipse.

Best regards

Bernard

[Non-text portions of this message have been removed]
• Dear Francois ... One explanation you can find is at http://www.pballew.net/isogon.html#lemhist where it is written Re: Also, why is it symbolized with K? At
Message 1 of 10 , Nov 12, 2007
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Dear Francois

> By the way, why this damned Lemoine point is always
> labelled K ?

One explanation you can find is at

http://www.pballew.net/isogon.html#lemhist

where it is written

Re: "Also, why is it symbolized with K?"
At the time, the choice of "K" came as the first
'unused' letter of the alphabet in Lemoine's scheme,
since the first letters A, B, C, ... were (and still
are) used for naming the vertices of a triangle, and
somehow accepted meaning:
F: [F]euerbach point (= center of the 9-point circle)
G: [G]ravity center (= Barycenter = Median point =
Centroid) H: [H]oehen(schnitt) punkt (= Orthocenter)
I: [I]ncenter
Therefore, within Lemoine's scheme, there was:
J: used for other purposes (especially, J_a, J_b, &
J_c) K: available

Best regards

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• Dear Bernard Thank you for your nice remarks and your references. I guessed that A B C had some name but I am losing my memory. Quelle tristesse! There is a
Message 1 of 10 , Nov 12, 2007
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Dear Bernard

I guessed that A'B'C' had some name but I am losing my memory.

Quelle tristesse!

There is a vast literature on inner Steiner ellipse foci and I don't
remember if this construction has already be given. No doubt yes but what is
interesting is the long choo-choo way I have followed to find it. Alas, I
have no room here to tell you the story.

In fact, the same choo-choo theorem gives as a special case the construction
of 2 isogonal conjugate points with given middle. I remember Paul Yiu gave
such a construction and maybe you or Jean-Pierre had sent me another one.
As for me, I have already found one also based on circular geometry but less
beautiful than this last one derived from choo-choo theory.

I can't resist to give you the general construction I found and you will
tell me if it is the same as Paul's one or yours.

So we want to construct 2 isogonal conjugate points F and F' wrt triangle
ABC, with given middle P.
They are the foci of the ABC-inner conic with center P.
It is easy to get the perspector J of the conic as the isotomic conjugate of
the anticomplement of P.
I hope I don't mix up here complement and anticomplement, funny names for
homotheties!
The cevian triangle U V W of P is the contact triangle of the conic with the
sides of ABC.

Now comes the construction strictly speaking:
Let T be the isogonal conjugate of P wrt ABC.
Line AT cuts again the (A V W) - circumcircle in A'.
Line BT cuts again the (B W U) - circumcircle in B'
Line CT cuts again the (C U V) - circumcircle in C'.

Here you can check that the 4 points A', B', C', T are on a same circle!

Let f be the direct circular map such that: f(A) = A', f(B) = B', f(C) = C'
Then f is of order 2, that is to say involutive.
The central point of f is P, f(P) = Infty and f(Infty) = P
The fixed points of f are the foci F and F' of the ABC-inner conic of center
P.

Hence we have 3 harmonic quadrangles (A, A', F, F'), (B, B', F, F'), (C, C',
F, F'), so giving an easy construction of these foci.
Friendly
Francois
PS
I like this construction for it is very symmetric though maybe a little
long!
Is it new ? I doubt it

On Nov 12, 2007 7:23 AM, Bernard Gibert <bg42@...> wrote:

> Dear Francois ,
>
>
> > So we start with our usual beloved triangle ABC, its circumcenter O
> > and its
> > Lemoine point K.
> > By the way, why this damned Lemoine point is always labelled K ?
>
> L is the de Longchamps point...
>
> > Let A', B', C' respectively the orthogonal projections of O onto the
> > symedians AK, BK, CK. ( They are on the Brocard circle.)
>
> A'B'C' is the second Brocard triangle.
>
> > Let f be the direct circular map such that: f(A) = A', f(B) = B', f
> > (C) =
> > C'.
> > Then f is of order 2, that is to say involutive.
>
> I suspect that your map is the one described in "orthocorrespondence
> and orthopivotal cubics" §5.
>
> f(M) is the product of an inversion and a reflection. It is also :
>
> - the center of the polar conic of M wrt the McCay or Kjp cubic,
>
> - the isoconjugate of M in the triangle formed by G and the circular
> points at infinity. In other words, the pole of M in the pencil of
> rectangular hyperbolas passing through the 4 foci of the Steiner
> inellipse.
>
> Best regards
>
> Bernard
>
> [Non-text portions of this message have been removed]
>
>
>

[Non-text portions of this message have been removed]
• Of course in so long a post, there are some typos always discovered just after sending the post ... Here one must read of course: The cevian triangle U V W of
Message 1 of 10 , Nov 12, 2007
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Of course in so long a post, there are some typos always discovered just
after sending the post

On Nov 12, 2007 5:07 PM, Francois Rideau <francois.rideau@...> wrote:

> Dear Bernard
>
> I guessed that A'B'C' had some name but I am losing my memory.
>
> Quelle tristesse!
>
> There is a vast literature on inner Steiner ellipse foci and I don't
> remember if this construction has already be given. No doubt yes but what is
> interesting is the long choo-choo way I have followed to find it. Alas, I
> have no room here to tell you the story.
>
> In fact, the same choo-choo theorem gives as a special case the
> construction of 2 isogonal conjugate points with given middle. I remember
> Paul Yiu gave such a construction and maybe you or Jean-Pierre had sent me
> another one.
> As for me, I have already found one also based on circular geometry but
> less beautiful than this last one derived from choo-choo theory.
>
> I can't resist to give you the general construction I found and you will
> tell me if it is the same as Paul's one or yours.
>
> So we want to construct 2 isogonal conjugate points F and F' wrt triangle
> ABC, with given middle P.
> They are the foci of the ABC-inner conic with center P.
> It is easy to get the perspector J of the conic as the isotomic conjugate
> of the anticomplement of P.
> I hope I don't mix up here complement and anticomplement, funny names for
> homotheties!
> The cevian triangle U V W of P is the contact triangle of the conic with
> the sides of ABC.

Here one must read of course:
The cevian triangle U V W of J, the perspector , is the contact triangle
of the conic with the sides of ABC......

>
>
> Now comes the construction strictly speaking:
> Let T be the isogonal conjugate of P wrt ABC.
> Line AT cuts again the (A V W) - circumcircle in A'.
> Line BT cuts again the (B W U) - circumcircle in B'
> Line CT cuts again the (C U V) - circumcircle in C'.
>
> Here you can check that the 4 points A', B', C', T are on a same circle!
>
> Let f be the direct circular map such that: f(A) = A', f(B) = B', f(C) =
> C'
> Then f is of order 2, that is to say involutive.
> The central point of f is P, f(P) = Infty and f(Infty) = P
> The fixed points of f are the foci F and F' of the ABC-inner conic of
> center P.
>
> Hence we have 3 harmonic quadrangles (A, A', F, F'), (B, B', F, F'), (C,
> C', F, F'), so giving an easy construction of these foci.
> Friendly
> Francois
> PS
> I like this construction for it is very symmetric though maybe a little
> long!
> Is it new ? I doubt it
>
>
>
>
>
>
>
> On Nov 12, 2007 7:23 AM, Bernard Gibert < bg42@...> wrote:
>
> > Dear Francois ,
> >
> >
> > > So we start with our usual beloved triangle ABC, its circumcenter O
> > > and its
> > > Lemoine point K.
> > > By the way, why this damned Lemoine point is always labelled K ?
> >
> > L is the de Longchamps point...
> >
> > > Let A', B', C' respectively the orthogonal projections of O onto the
> > > symedians AK, BK, CK. ( They are on the Brocard circle.)
> >
> > A'B'C' is the second Brocard triangle.
> >
> > > Let f be the direct circular map such that: f(A) = A', f(B) = B', f
> > > (C) =
> > > C'.
> > > Then f is of order 2, that is to say involutive.
> >
> > I suspect that your map is the one described in "orthocorrespondence
> > and orthopivotal cubics" §5.
> >
> > f(M) is the product of an inversion and a reflection. It is also :
> >
> > - the center of the polar conic of M wrt the McCay or Kjp cubic,
> >
> > - the isoconjugate of M in the triangle formed by G and the circular
> > points at infinity. In other words, the pole of M in the pencil of
> > rectangular hyperbolas passing through the 4 foci of the Steiner
> > inellipse.
> >
> > Best regards
> >
> > Bernard
> >
> > [Non-text portions of this message have been removed]
> >
> >
> >
>
>

[Non-text portions of this message have been removed]
• Dear Nikos Thank you for the references. Of course I knew the name of Grebe for it appears in the F.G-M with his theorem , n° 2369, page 1181, followed by a
Message 1 of 10 , Nov 12, 2007
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Dear Nikos
Thank you for the references.
Of course I knew the name of Grebe for it appears in the F.G-M with his
theorem , n� 2369, page 1181, followed by a short article on Grebe's life.
That's the label K I did not understand and now , I know why thanks to you!
Friendly
Francois

>
> Dear Francois
>
>
> > By the way, why this damned Lemoine point is always
> > labelled K ?
>
> One explanation you can find is at
>
> http://www.pballew.net/isogon.html#lemhist
>
> where it is written
>
> Re: "Also, why is it symbolized with K?"
> At the time, the choice of "K" came as the first
> 'unused' letter of the alphabet in Lemoine's scheme,
> since the first letters A, B, C, ... were (and still
> are) used for naming the vertices of a triangle, and
> the letters F, G, H, I had already a standard and
> somehow accepted meaning:
> F: [F]euerbach point (= center of the 9-point circle)
> G: [G]ravity center (= Barycenter = Median point =
> Centroid) H: [H]oehen(schnitt) punkt (= Orthocenter)
> I: [I]ncenter
> Therefore, within Lemoine's scheme, there was:
> J: used for other purposes (especially, J_a, J_b, &
> J_c) K: available
>
> Best regards
>
>
>
>
> ___________________________________________________________
> �������������� Yahoo!;
> ���������� �� ���������� �������� (spam); �� Yahoo! Mail
> �������� ��� �������� ������ ��������� ���� ��� �����������
>
>
>
>
>
>
>
>
>

[Non-text portions of this message have been removed]
• Dear friends I want to add some remarks on this foci construction. To get them, we have the choice between 3 harmonic quadrangles (A, A , F, F ); (B, B ; F,
Message 1 of 10 , Nov 14, 2007
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Dear friends
I want to add some remarks on this foci construction.
To get them, we have the choice between 3 harmonic quadrangles (A, A', F,
F'); (B, B'; F, F'); (C, C', F, F')
So for example we can forget vertices B and C and only look at triangle A V
W.

In other words, as a by-product, we get the construction of the foci of the
conic <Gamma> of center O tangent in V and W respectively to the sides AV
and AW.

We know that line AO is the (A V W ) -median through A.
We draw the symedian of A V W through A.
This symedian cuts again the (A V W) -circumcircle in A'.
And if F and F' are the foci of <Gamma>, the quadrangle (A, A', F, F') is
harmonic and so we get an easy construction of F and F'.

In the special case where <Gamma> is an ellipse and OV and OW conjugate
diameters, it would be interesting to compare this construction with the
very famous one
giving the foci of an ellipse knowing 2 conjugate diameters.
I notice both constructions use a harmonic quadrangle, so there must be some
relationship between them.

If the triangle A V W is given, the locus of the center of the conic
tangents in V and W respectively to the lines AV and AW is the median
through O as I have said it.
Besides these conics are in a pencil both linear and tangential.

As for the foci locus, it is some strophoid through V and W and the middle
of AA', with a double point in A.
Moreover this middle of AA' is the focus of the parabola in the pencil.
I forgot the construction of this sthrophoid asymptot!
So help me to remember it!

Friendly
Francois
PS
Of course, I am sure that this by-product construction can be found
somewhere in some old books and maybe has already been given here in the
past.

[Non-text portions of this message have been removed]
• Dear friends Now, I remember how to get this damn asymptot and the classic generation of this locus as a strophoid. As I guessed, I has found again this
Message 1 of 10 , Nov 14, 2007
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Dear friends
Now, I remember how to get this damn asymptot and the classic generation of
this locus as a strophoid.
As I guessed, I has found again this construction as an exercise on circular
geometry in the old book from Iliovici and Robert.
Even the strophoid locus was given and its invariance by the transposition
with fixed points A and A' proved.
And all that stuff in an A-level book, it was another world!l
Friendly
Francois

On Nov 14, 2007 1:57 PM, Francois Rideau <francois.rideau@...> wrote:

> Dear friends
> I want to add some remarks on this foci construction.
> To get them, we have the choice between 3 harmonic quadrangles (A, A', F,
> F'); (B, B'; F, F'); (C, C', F, F')
> So for example we can forget vertices B and C and only look at triangle A
> V W.
>
> In other words, as a by-product, we get the construction of the foci of
> the conic <Gamma> of center O tangent in V and W respectively to the sides
> AV and AW.
>
> We know that line AO is the (A V W ) -median through A.
> We draw the symedian of A V W through A.
> This symedian cuts again the (A V W) -circumcircle in A'.
> And if F and F' are the foci of <Gamma>, the quadrangle (A, A', F, F') is
> harmonic and so we get an easy construction of F and F'.
>
> In the special case where <Gamma> is an ellipse and OV and OW conjugate
> diameters, it would be interesting to compare this construction with the
> very famous one
> giving the foci of an ellipse knowing 2 conjugate diameters.
> I notice both constructions use a harmonic quadrangle, so there must be
> some relationship between them.
>
> If the triangle A V W is given, the locus of the center of the conic
> tangents in V and W respectively to the lines AV and AW is the median
> through O as I have said it.
> Besides these conics are in a pencil both linear and tangential.
>
> As for the foci locus, it is some strophoid through V and W and the middle
> of AA', with a double point in A.
> Moreover this middle of AA' is the focus of the parabola in the pencil.
> I forgot the construction of this sthrophoid asymptot!
> So help me to remember it!
>
> Friendly
> Francois
> PS
> Of course, I am sure that this by-product construction can be found
> somewhere in some old books and maybe has already been given here in the
> past.
>

[Non-text portions of this message have been removed]
• Dear Francois, You have mentioned the book of Iliovici and Robert. Can you please tell some more about the authors and their book? What is an A-level book?
Message 1 of 10 , Nov 14, 2007
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Dear Francois,

You have mentioned the book of Iliovici and Robert. Can you please tell
some more about the authors and their book? What is an A-level book?

Sincerely, Jeff
• Dear Jeff By A-level, I intend to say last year of high school level , that is to say baccalaureat year in french, but maybe my translation was wrong or
Message 1 of 10 , Nov 15, 2007
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Dear Jeff
By A-level, I intend to say "last year of high school level", that is to say
"baccalaureat year " in french, but maybe my translation was wrong or only
valid in UK?
Iliovici and Robert is a very good book on geometry, published in 1937,
destined for pupils on their last year of high school.
At this time, you must have a very good level in geometry to enter
university or some engineer school.
Now a pocket adding machine is enough in case you forget the additive
monoid N structure!
Multiplication is not needed!
The book was centered on plane transformations, isometries, similitudes,
inversions and on conics defined by foci and directrix. But the last chapter
was on circular geometry, even if it was not in the program.
You cannot find it anywhere even here in France. So I am happy to have it!
Friendly
Francois

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