#### Abstract

We group triangles into families based on three parameters: the distance between the circumcenter O and the centroid G, the circumradius, and the measure of angle Ð GOAwhere A is one vertex. We focus on the family of triangles which allows Ð GOA to vary and fixes the other two parameters. By construction, this grouping produces triangles which share the same Euler line. Perhaps unexpectedly, if we examine the family's locus of a triangle center known as the symmedian point, we find that it always forms an arc of a circle centered at a specified point on the Euler line.

#### Author Bio

Originally from Milwaukee, Wisconsin (www.mil-mil.net), Adam Carr is headed off to Corfu, Greece after graduation from Carleton in 2007. There, he will manage a volunteer organization for an organic kiwi (-fruit) farm and pony (-animal) stable. Upon return, he looks to enter a two year teaching (-human) program. He and his three colleagues wrote "Natural Families of Triangles I: Parametrizing Triangle Space" and "Natural Families of Triangles II: A Locus of Symmedian Points" as part of their senior thesis.

Julia Fisher received her undergraduate degree in mathematics from Carleton College in 2007. She will be spending the next two years teaching high school mathematics with Teach For America in the Rio Grande Valley in southern Texas. Fisher is particularly interested in geometry, real analysis, and large LaTeX manuals. She and her three colleagues wrote "Natural Families of Triangles I: Parametrizing Triangle Space" and "Natural Families of Triangles II: A Locus of Symmedian Points" as part of their senior thesis.

Andrew Roberts, Carleton �07, plans to begin teaching high school mathematics in Bay St. Louis, MS in the fall of 2007. Eventually, he plans to attend graduate school for a Ph.D. in math. If he were a Disney princess, he�d be Pocahontas. He and his three colleagues wrote "Natural Families of Triangles I: Parametrizing Triangle Space" and "Natural Families of Triangles II: A Locus of Symmedian Points" as part of their senior thesis.

Peng (David) Xu received his undergraduate degree in mathematics and economics from Carleton College in 2007. He will be working at UBS Investment Bank next year. Eventually, he plans to do finance in the Greater China region. During a management consulting interview once, David convinced his interviewer that abstract algebra is pertinent to consulting. He and his three colleagues wrote "Natural Families of Triangles I: Parametrizing Triangle Space" and "Natural Families of Triangles II: A Locus of Symmedian Points" as part of their senior thesis.

#### Recommended Citation

Carr, Adam; Fisher, Julia; Roberts, Andrew; and Xu, Peng (David)
(2008)
"Natural Families of Triangles II: A Locus of Symmedian Points,"
*Rose-Hulman Undergraduate Mathematics Journal*: Vol. 9
:
Iss.
1
, Article 3.

Available at:
https://scholar.rose-hulman.edu/rhumj/vol9/iss1/3

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