Volume 13, No. 4

March, 2004

by Jaime Wren

Triangle geometry was developed so much that discoveries began to slow. However over the last few decades Dr. Clark Kimberling, a math professor at the University of Evansville has brought a lot of attention back to triangle geometry.  Using trilinear or barycentric coordinates, a form of homogeneous coordinates (an invention from Möbius,)  Professor Kimberling quickly turned triangle loci, triangles, circles and centers into a massive index of linear algebra matrices.  With the input of other interested mathematicians, Kimberly has, as of this February, cataloged over 1100 triangle centers.  With enough time spent on Geometry Sketchpad anyone could potentially find an undiscovered locus or center of the triangle. As an interesting side note, the 384^th point in his encyclopedia was sent to him via email in 1998, by John Conway, Kimberling calls this point the Conway Point.
I encourage interested readers to view Kimberling's "Encyclopedia of Triangle Centers," he maintains it online at http://faculty.evansville.edu/ck6/encyclopedia/ETC.html.
Eric W. Weisstein. "Kimberling Center." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/KimberlingCenter.html
Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-167, 1994.

The Greeks studied four notable points of the triangle the centroid, incenter, circumcenter, and orthocenter.  The centroid is also called the "center of gravity" and is at the intersection of the medians of the triangle.  The incenter, the point equidistant to each of the sides of the triangle, is at the intersection of the angle bisectors.  The circumcenter is the point equidistant to the vertices of the triangle.  Finally, the orthocenter is at the intersection of the altitudes, where an altitude is defined as a line segment dropped from a triangle point to the opposite side perpendicularly.
Two thousand years after the Greeks discovered and studied these centers, Howard Eves attached an equilateral triangle onto each side of a triangle and discovered another center to the triangle, now called the Fermat point.  With the discovery of a fifth triangle center, mathematicians naturally asked if there existed other significant triangle centers.  With time, a plethora of triangle properties, loci, and centers were discovered, to name a few: the nine-point center, the Gergonne point, the Lemoine point, the Speiker point, and the Clawson point.