## Re: [EMHL] Asymptotes of Kiepert hyperbola

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• Dear Paul, ... Do we know any notable points lying on this circle with diameter the Fermats? APH
Message 1 of 4 , Aug 21, 2000
Dear Paul,

You wrote:

> Talking about the center of the Kiepert hyperbola, I just recalled
>that it is also the midpoint of the Fermat points.

Do we know any notable points lying on this circle with diameter
the Fermats?

APH
• Let s continue this interesting discussion ! On Wed, 29 Dec 1999, Bernard Gibert wrote: ... I have not this remarkable book and I don t know where could one
Message 2 of 4 , Aug 30, 2000
Let's continue this interesting discussion !

On Wed, 29 Dec 1999, Bernard Gibert wrote:

[APH]:
>>J. R. Musselman published in the AMM 45 (1938) 482, two problems
>>on Kiepert (and Jerabek) hyperbola:
>>
>>The pedal circle of the centroid G of a triangle A_1A_2A_3 passes through
>>the centers of the hyperbolas of Kiepert and Jerabek.
>
> Those two points are called 1st and 2nd Schroeter points : they both lie on
> the 9pt circle. (Schroeter 1865)
> They have quite a lot of interesting properties.
> Among them, the 2nd point is the fourth intersection point of the 9pt circle
> and the ellipse K.
>
> May I recommend the excellent book (in French) by Collet & Griso called " le
> cercle d'Euler " where a whole chapter is dedicated to them.

I have not this remarkable book and I don't know where could one obtain a
copy from (great French on-line bookstores, like chapitre, have no copies).

Scrhoeter construction of Jerabek/Kiepert H. centers is this - if I remember
correctly the construction I read once somewhere:

Let A'B'C', A"B"C" be the orthic, medial triangles of a triangle ABC. Denote:
B'C' /\ B"C" := A*
C'A' /\ C"A" := B*
A'B' /\ A"B" := C*

Then:

A'A*, B'B* , C'C* concur at Jerabek H. center
A"A*, B"B* , C"C* concur at Kiepert H. center

I believe that Scrhoeter studied the general case of two arbitrary (?) points
P,Q, and their cevian triangles A'B'C', A"B"C".
Are in general the triads of lines (A'A*, B'B* , C'C*), (A"A*, B"B* , C"C*)
concurrent? (A*,B*,C* as defined above)
Also, where the points of concurrences are lying on ? (in which pedal/cevian
circles?)
[If P = H, Q = I, then do we obtain Feuerbach H. center?]

Antreas
• ... Yes, they are. I proved it by Menelaus/Ceva Theorems. The theorem is: Let P, Q be two points on the plane of ABC, and A B C , A B C their cevian
Message 3 of 4 , Aug 31, 2000
I wrote:

>Let's continue this interesting discussion !
>
>On Wed, 29 Dec 1999, Bernard Gibert wrote:
>
>[APH]:
>>>J. R. Musselman published in the AMM 45 (1938) 482, two problems
>>>on Kiepert (and Jerabek) hyperbola:
>>>
>>>The pedal circle of the centroid G of a triangle A_1A_2A_3 passes through
>>>the centers of the hyperbolas of Kiepert and Jerabek.
>>
>> Those two points are called 1st and 2nd Schroeter points : they both lie on
>> the 9pt circle. (Schroeter 1865)
>> They have quite a lot of interesting properties.
>> Among them, the 2nd point is the fourth intersection point of the 9pt circle
>> and the ellipse K.
>>
>> May I recommend the excellent book (in French) by Collet & Griso called " le
>> cercle d'Euler " where a whole chapter is dedicated to them.
>
>I have not this remarkable book and I don't know where could one obtain a
>copy from (great French on-line bookstores, like chapitre, have no copies).
>
>Scrhoeter construction of Jerabek/Kiepert H. centers is this - if I remember
>correctly the construction I read once somewhere:
>
>Let A'B'C', A"B"C" be the orthic, medial triangles of a triangle ABC. Denote:
>B'C' /\ B"C" := A*
>C'A' /\ C"A" := B*
>A'B' /\ A"B" := C*
>
>Then:
>
>A'A*, B'B* , C'C* concur at Jerabek H. center
>A"A*, B"B* , C"C* concur at Kiepert H. center
>
>I believe that Scrhoeter studied the general case of two arbitrary (?) points
>P,Q, and their cevian triangles A'B'C', A"B"C".
>Are in general the triads of lines (A'A*, B'B* , C'C*), (A"A*, B"B* , C"C*)
>concurrent? (A*,B*,C* as defined above)

Yes, they are. I proved it by Menelaus/Ceva Theorems.

The theorem is:
Let P, Q be two points on the plane of ABC, and A'B'C', A"B"C" their cevian
triangles. Denote:
B'C' /\ B"C" := A*
C'A' /\ C"A" := B*
A'B' /\ A"B" := C*

Then:
A'A*, B'B* , C'C* concur.
A"A*, B"B* , C"C* concur.
(or in other words: The triangle A*B*C* is in perspective with both
triangles A'B'C', A"B"C")

[The proof in my next posting, with subject line: Another Fruitful Theorem]

Does anyone know a reference?

Had Schroeter discovered the theorem, or just applied it for P = H, Q = G,
and found that the concurrence points are the centers of Jerabek/Kiepert H.?

Antreas

>Also, where the points of concurrences are lying on ? (in which pedal/cevian
>circles?)
>[If P = H, Q = I, then do we obtain Feuerbach H. center?]
>
>Antreas
• ... De : À : Envoyé : vendredi 1 septembre 2000 02:10 Objet : Re: [EMHL] Asymptotes of Kiepert hyperbola Dear
Message 4 of 4 , Sep 2, 2000
----- Message d'origine -----
De : <xpolakis@...>
À : <Hyacinthos@egroups.com>
Envoyé : vendredi 1 septembre 2000 02:10
Objet : Re: [EMHL] Asymptotes of Kiepert hyperbola

Dear Antreas and other Hyacinthists,
Antreas wrote :
> The theorem is:
> Let P, Q be two points on the plane of ABC, and A'B'C', A"B"C" their
cevian
> triangles. Denote:
> B'C' /\ B"C" := A*
> C'A' /\ C"A" := B*
> A'B' /\ A"B" := C*
>
> Then:
> A'A*, B'B* , C'C* concur.
> A"A*, B"B* , C"C* concur.
> (or in other words: The triangle A*B*C* is in perspective with both
> triangles A'B'C', A"B"C")

We can add that A', B', C', A", B", C" and the two perspectors lie on a same
conic;
if Dp and Dq are the trilinear polars of P and Q, this conic is the
locus of the poles of Dp wrt the circumconics going through Q and the locus
of the poles of Dq wrt the circumconics going through P.
This is just a projective generalization of the particular case P = G, Q = H
where the conic is the NPC and (the poles of Dp wrt the circumconics going
through Q = the centers of the rectangular circumhyperbolas)

If P =(x1,y1,z1) and Q = (x2,y2,z2), the conic is
x x/x1/x2 - (1/y1/z2 + 1/y2/z1) y z + circular = 0.

Friendly from France. Jean-Pierre
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