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
>> 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 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,
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
>> > 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
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 \ /
This is the "Jacobi" case.
>> Apex-angle (al,be,ga)-Naps defined like this
>> \al /\ ga/
>> \ / \ /
>> \ / \ /
>> \ / \ /
>> \ /
>> \ /
>> 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.