What is Extraversion in Triangle Geometry?

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SUMMARY

Extraversion in triangle geometry, a term coined by John Conway, refers to the transformation of triangle ABC by smoothly moving vertices A and C through each other. This process results in the incircle exchanging positions with the b-excircle, demonstrating a relationship between their properties. The concept is discussed in detail in Conway's MathFest talk and is further explored in the Steiner-Lehmus angle-bisector theorem published in 2015. For visual learners, animations illustrating this transformation are available online.

PREREQUISITES
  • Understanding of triangle geometry concepts, including incircles and excircles.
  • Familiarity with the properties of triangle vertices and their relationships.
  • Basic knowledge of algebraic results related to geometric transformations.
  • Experience with mathematical terminology and theorems in geometry.
NEXT STEPS
  • Research the Steiner-Lehmus angle-bisector theorem for deeper insights into triangle properties.
  • Explore John Conway's work on triangle geometry and his contributions to mathematical terminology.
  • Examine animations and visual representations of extraversion in triangle geometry.
  • Learn about the implications of vertex movement on triangle properties and related theorems.
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Mathematicians, geometry enthusiasts, educators, and students interested in advanced triangle geometry concepts and transformations.

Cheesycheese213
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I came across something called "extraversion", which in that case, was swapping 2 verticies to move the triangle created by it's trisectors outside of the original triangle. I found it in a blog post:
http://blog.zacharyabel.com/2012/03/many-morley-triangles/
but I can't seem to find any other reference to it other than a different blog post by the same author. Is there any article or place I can learn more about it, and if not, is there a different word that describes it? Thanks!
 
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Extraversion: Extraversion is John Conway’s word for the study of what happens to theorems in triangle geometry as you smoothly move two vertices A and C of a triangle ABC through each other. (See a nice animation athttp://bit.ly/1gYNA82.) This movement causes the incircle (or inscribed circle) of the original ABC to change places with the b-excircle (see http://bit.ly/1HY74zi for a definition of excircle). And for any algebraic result about the incircle or incenter, a corresponding result holds for the b-excircle or excenter as long as you change the sign of b. (The incenter and excenter are the centers of the incircle and excircle, respectively.)

“There’s a pun, of course,” Conway said of extraversion in his MathFest talk (which followed Guy’s), “since I invented the term.” Extraversion involves “extraverting” a triangle or turning it inside out, Conway explained, but it also produces “extra versions” of various entities.

This might be the first published use:
The Steiner-Lehmus angle-bisector theorem
John Conway, Alex Ryba
https://doi.org/10.1017/S0025557200001236
Published online: 23 January 2015, pp. 193-203
 
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