- Dear John Conway,

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

Thanks for the reply.

>> Yes, at least to me. It's much more general than

A little historical digression. I think this is what you

>> 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).

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)

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

>> also form a triangle in perspective with ABC,

points?

Sincerely,

Darij Grinberg - 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 - On Thu, 12 Jun 2003, Darij Grinberg wrote:

> Dear John Conway,

I should explain my long absence. It was caused by the

>

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

> Thanks for the reply.

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

I have never used any of those names, but knew it was at

> 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.

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,

Many thanks again. You wrote:

>> I should explain my long absence. It was caused by the

archived

>> 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

>> those messages, and plan to go through them "some day".] However,

I wish you good luck with following up the emails.

>> 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.

Let me tell you how I joined Hyacinthos in December 2002

and had to read through all messages from #1 to #6122. First,

I had to download all them in HTML files partitioned with 15

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

Another source, may be more reliable:

>> > 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.

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

Hope you won't mind me digressing again. Let triangles

>> 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.

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

I think such triangles are not perspective with ABC. In fact,

>> 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.

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

But they are not defined by one angle! The circumcircles

>>

>> ___________________

>> \al /\ ga/

>> \ / \ /

>> \ / \ /

>> \ / \ /

>> ---------

>> \ /

>> \ /

>> \be/

>>

>> have circumcircles whose radical center is the other type.

are uniquely defined, but not the triangles themselves.

>> The two types are mutually isogonal. If al+be+ga = pi, then

Now it seems that we are speaking of different things, if not

>> the circumcircles just mentioned concur in the appropriate pivot.

something is wrong.

Sincerely,

Darij Grinberg