A014498 and JHC's extraversion

John Conway conway at math.Princeton.EDU
Wed Jun 7 01:37:48 CEST 2000

On Wed, 7 Jun 2000, Antreas P. Hatzipolakis wrote:

> Someone asked me what Conway's extraversion is.
> In my understanding, it is a metatheorem:
>   When you come across a theorem on the incircle/incenter/angle int.
>   bisectors, expect an analogous theorem on the excircles/excenters/
>   angle ext.bisectors.

   An "expectation" wouldn't deserve to be called 
      either a "theorem" or a "metatheorem".

   "Extraversion" is my name for the study of what happens
to theorems in triangle geometry as you smoothly move the
vertices of your triangle about.  In particular, if you
move two vertices   A  and  C  through each other:

        B                       B
 Then  / \                     / \
      /   \     becomes       /   \
     /     \                 /     \
    C->   <-A             <-A       C->

and the incircle changes places with the b-excircle.

    This entails the certainty (not merely an expectation)
that any algebraic result about the incircle or incenter
will yield a corresponding one about the b-excircle or excenter
by changing the sign of "b" (keeping "a" and "c" fixed).

    So Antreas is partly right (many theorems about the incircle
are ultimately algebraic, and so will certainly have analogs 
for the excircles) and partly wrong (for instance because
extraversion fails when inequalities - e.g.  b > 0  - are involved, 
and because it's far more general than just the expectation
he mentioned).  

    Extraversion is a particular case of what's called "monodromy"
in the technical language of modern algebraic geometry.  But because
the rest of that language is rather frightening (and unnecessary)
for beginners, I thought a special word was justified.  The word
"extraversion" is appropriate both because it involves "extraverting"
the triangle (turning it inside out) and because it leads to 
"extra versions" of various concepts (for instance the three
excenters are "extra versions" of the incenter).

    John Conway

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