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• ## Re: [EMHL] Inverting Kiepert triangles

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• Dear John Conway, I am very glad to see you posting at Hyacinthos again! Thanks for the reply. ... A little historical digression. I think this is what you and
Message 1 of 6 , Jun 12, 2003
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Dear John Conway,

I am very glad to see you posting at Hyacinthos again!

>> Yes, at least to me. It's much more general than
>> the Kiepert situation. Erect what I call
>> (alpha,beta,gamma)-Napoleons on the edges, namely
>> triangles whose base angles are alpha at A, beta at B,
>> gamma at C, then their apices form a triangle that's
>> in perspective with ABC at a point P(alpha,beta,gamma).

A little historical digression. I think this is what you
and Antreas call "Traditional Theorem" or "Isogonal Theorem"
or "de Villiers Theorem". In fact, it is very old - for
instance, see

William Allen Whitworth, "Trilinear Coordinates",
Cambridge 1986.

(This book is accessible through
http://library5.library.cornell.edu/math_W.html
.)

On page 57, the exercise (41) reads:

"On the three sides of a triangle ABC triangles PBC,
QCA, RAB are described so that the angles QAC, RAB
are equal, the angles RBA, PBC are equal, and the
angles PCB, QCA are equal; prove that the straight
lines, AP, BQ, CR pass through one point."

In one paper, Armin Saam calls this fact "Jacobi
Theorem", however he doesn't say which Jacobi this is
and where the fact was published first.

>> The inverses of the apices (in the circumcircle)
>> also form a triangle in perspective with ABC,

My sketches don't confirm that! Have we confused some
points?

Sincerely,
Darij Grinberg
• ... ^^^^ 1886. With my apologies for the typo, Sincerely, Darij Grinberg
Message 2 of 6 , Jun 13, 2003
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I wrote:

>> A little historical digression. I think this is what you
>> and Antreas call "Traditional Theorem" or "Isogonal Theorem"
>> or "de Villiers Theorem". In fact, it is very old - for
>> instance, see
>>
>> William Allen Whitworth, "Trilinear Coordinates",
>> Cambridge 1986.
^^^^
1886.

With my apologies for the typo,
Sincerely,
Darij Grinberg
• ... I should explain my long absence. It was caused by the large backlog that built up, initially because I went away from Princeton for two weeks sometime
Message 3 of 6 , Jun 13, 2003
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On Thu, 12 Jun 2003, Darij Grinberg wrote:

> Dear John Conway,
>
> I am very glad to see you posting at Hyacinthos again!

I should explain my long absence. It was caused by the
large backlog that built up, initially because I went away
from Princeton for two weeks sometime before Christmas, and
then got larger and larger because I couldn't face handling
it.

A few weeks ago, I solved this problem by just "throwing away"
more than 14,000 messages, and starting again. [I actually archived
those messages, and plan to go through them "some day".] However,
I find I'm now getting between 100 and 200 every day (most of
which are junk), and so am still having difficulty keeping up
to date.

At the moment, the way I'm doing this is just by deleting
most of them without reading, even when I know I might be
interested in them, as with the Hyacinthos ones. But I plan
to find some automatic way of getting rid of a goodly number
and will them start taking an interest in Hyacinthos again.

> A little historical digression. I think this is what you
> and Antreas call "Traditional Theorem" or "Isogonal Theorem"
> or "de Villiers Theorem". In fact, it is very old - for
> instance, see
>
> William Allen Whitworth, "Trilinear Coordinates",
> Cambridge 1986.

I have never used any of those names, but knew it was at
least a century old. Somehow I doubt "Jacobi", though.

I think I said a few things wrong. In our book, Steve
and I use this theorem to define the generalized pivot points
P^(alpha,beta,gamma) and P_(alpha,beta,gamma), which become
ordinary pivot points when alpha + beta + gamma = pi.
perhaps some of the assertions only hold in that case.

I regard the theorem as a fundamental one in view of
they importance of these concepts. I think it's foolish
to try to name it after someone (especially a recent someone)
for the usual reasons - any such attribution will almost
certainly be wrong, and in any case it wouldn't help to
recall the theorem. So let's call it something like
"the generalized pivot theorem".

Actually, there are two, corresponding to the two types
of pivot. The base-angle (al,be,ga)-Napoleons defined like this:

___________
ga / \al
/ B \
/ \
\be / \ be/
\ / C A \ /
------------
\al ga/
\ /

have apices in perspective with ABC, the perspector being the
first type of generalized pivot. Apex-angle (al,be,ga)-Naps
defined like this

___________________
\al /\ ga/
\ / \ /
\ / \ /
\ / \ /
---------
\ /
\ /
\be/

have circumcircles whose radical center is the other type.
The two types are mutually isogonal. If al+be+ga = pi, then
the circumcircles just mentioned concur in the appropriate pivot.

John Conway
• Dear John Conway, ... archived ... I wish you good luck with following up the emails. Let me tell you how I joined Hyacinthos in December 2002 and had to read
Message 4 of 6 , Jun 13, 2003
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Dear John Conway,

Many thanks again. You wrote:

>> I should explain my long absence. It was caused by the
>> large backlog that built up, initially because I went away
>> from Princeton for two weeks sometime before Christmas, and
>> then got larger and larger because I couldn't face handling
>> it.
>>
>> A few weeks ago, I solved this problem by just "throwing away"
>> more than 14,000 messages, and starting again. [I actually
archived
>> those messages, and plan to go through them "some day".] However,
>> I find I'm now getting between 100 and 200 every day (most of
>> which are junk), and so am still having difficulty keeping up
>> to date.
>>
>> At the moment, the way I'm doing this is just by deleting
>> most of them without reading, even when I know I might be
>> interested in them, as with the Hyacinthos ones. But I plan
>> to find some automatic way of getting rid of a goodly number
>> and will them start taking an interest in Hyacinthos again.

I wish you good luck with following up the emails.

Let me tell you how I joined Hyacinthos in December 2002
and had to read through all messages from #1 to #6122. First,
messages in each file. However, copying took five days, but
reading took some months. I also tried to reprove some of the
theorems I found in Hyacinthos, what was a quite good way of
appreciating them.

>> > A little historical digression. I think this is what you
>> > and Antreas call "Traditional Theorem" or "Isogonal Theorem"
>> > or "de Villiers Theorem". In fact, it is very old - for
>> > instance, see
>> >
>> > William Allen Whitworth, "Trilinear Coordinates",
>> > Cambridge 1986.
>>
>> I have never used any of those names, but knew it was at
>> least a century old. Somehow I doubt "Jacobi", though.

Another source, may be more reliable:

Peter Baptist, "Die Entwicklung der neueren
Dreiecksgeometrie", Mannheim-Leipzig-Wien-Zürich 1992.

This book also manifests Jacobi as the first discoverer (alas,
I have only a copy of the pages with the geometry and don't
know the initials of Jacobi). Jacobi presented his theorem in
his schoolbook.

Of course, this is not a reason to name it after Jacobi, if
there is a good alternative term.

>> I think I said a few things wrong. In our book, Steve
>> and I use this theorem to define the generalized pivot points
>> P^(alpha,beta,gamma) and P_(alpha,beta,gamma), which become
>> ordinary pivot points when alpha + beta + gamma = pi.
>> perhaps some of the assertions only hold in that case.

Hope you won't mind me digressing again. Let triangles
BA'C, CB'A and AC'B are constructed on BC, CA, AB so that

angle A'BC = angle BCA' = alpha;
angle B'CA = angle CAB' = beta;
angle C'AB = angle ABC' = gamma.

Here, I consider DIRECTED ANGLES MODULO 180°. (These are
THE angles of circle geometry.)

Now, I usually call triangle A'B'C' the
(alpha, beta, gamma)-Jacobi triangle. If

alpha + beta + gamma = 0°,

then, I also call triangle A'B'C' a special Jacobi
triangle. (I formerly called it special Schaal triangle -
forget this nonsense name.) While for any Jacobi triangle,
the lines AA', BB', CC' concur, for any special Jacobi
triangle, the point of concurrence also lies on the
circles A'BC, B'CA and C'AB.

However, neither for all Jacobi triangles nor for all
special Jacobi triangles the inverses in the circumcircle
form a triangle perspective to ABC. Actually, they do for
all Kiepert triangles, which is due to the fact that the
vertices of a Kiepert triangle lie on the perpendicular
bisectors of the sides of ABC.

>> I regard the theorem as a fundamental one in view of
>> they importance of these concepts. I think it's foolish
>> to try to name it after someone (especially a recent someone)
>> for the usual reasons - any such attribution will almost
>> certainly be wrong, and in any case it wouldn't help to
>> recall the theorem. So let's call it something like
>> "the generalized pivot theorem".
>>
>> Actually, there are two, corresponding to the two types
>> of pivot. The base-angle (al,be,ga)-Napoleons defined like this:
>>
>>
>> ___________
>> ga / \al
>> / B \
>> / \
>> \be / \ be/
>> \ / C A \ /
>> ------------
>> \al ga/
>> \ /
>>
>> have apices in perspective with ABC, the perspector being the
>> first type of generalized pivot.

I think such triangles are not perspective with ABC. In fact,
if they were perspective, we would have

cot A + cot be cot B + cot ga cot C + cot al
-------------- * -------------- * -------------- = 1,
cot A + cot ga cot B + cot al cot C + cot be

what doesn't seem to be correct.

Perhaps the angles must be positioned another way round:

___________
be / \be
/ B \
/ \
\ga / \ al/
\ / C A \ /
------------
\ga al/
\ /

This is the "Jacobi" case.

>> Apex-angle (al,be,ga)-Naps defined like this
>>
>> ___________________
>> \al /\ ga/
>> \ / \ /
>> \ / \ /
>> \ / \ /
>> ---------
>> \ /
>> \ /
>> \be/
>>
>> have circumcircles whose radical center is the other type.

But they are not defined by one angle! The circumcircles
are uniquely defined, but not the triangles themselves.

>> The two types are mutually isogonal. If al+be+ga = pi, then
>> the circumcircles just mentioned concur in the appropriate pivot.

Now it seems that we are speaking of different things, if not
something is wrong.

Sincerely,
Darij Grinberg
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