Encyclopedia of Triangle Centers

Clark Kimberling, a professor of mathematics at the University of Evansville, Evansville, IN, maintains a list of triangle centers. They are labeled by number as (X₁), (X₂), etc. The number of centers associated with triangles is over 5000.

The first ten triangle centers are:

1. (X₁) - the incenter, the intersection of interior angle bisectors
2. (X₂) - the centroid, the intersection of medians
3. (X₃) - the circumcenter, the intersection of perpendicular bisectors of sides
4. (X₄) - the orthocenter, the intersection of altitudes
5. (X₅) - the nine-point center, the center of the nine-point circle
6. (X₆) - the symmedian point, the intersection of the three symmedians
7. (X₇) - the Gergonne point, the symmedian point of the contact triangle
8. (X₈) - the Nagel point, the intersection of lines from a vertex to its semiperimeter point
9. (X₉) - the mittelpunkt, the symmedian point of the triangle formed by the centers of the three excircles
10. (X₁₀) - the Spielker center, the center of the Spielker circle

A few others that are less commonly mentioned, but pop up in many geometry discussions, include:

1. (X₁₁) - the Feuerbach point
2. (X₁₃) - the Fermat point
3. (X₂₀) - the de Longchamps point
4. (X₂₁) - the Schiffler point
5. (X₃₉) - the Brocard midpoint

What makes triangle geometry so much fun is discovering that some of these centers fall on the same line. When three or more points have constant distance ratios, that's even more amazing. Two sets of points that exhibit this property are the Euler Line and a line that contains the Spieker point. The Euler line contains the centroid (X₂), the circumcenter (X₃), and the orthocenter (X₄). The distance between the centroid and the orthocenter is always twice that of the distance between the centroid and the circumcenter. The great Swiss mathematician Leonhard Euler discovered this wholly unexpected relationship in 1765. The other set of collinear points is the incenter X₁), the centroid, the Nagel (X₈) point and the Spieker point (X₁₀). The ratio of distances between points is also fixed: |(X₈)(X₁₀)|:|(X₈)(X₂)|:|(X₈)(X₁)| = 3:4:6.

The Encyclopedia of Triangle Geometry is commonly abbreviated as "ETC" by geometers.

Internet References

1. Encyclopedia of Triangle Centers. A major resource for geometers
2. Wikipedia, "Encyclopedia of Triangle Centers
3. Clark Kimberling
4. Triangle Geometers. Mathematicians, both professional and laymen, who specialized in triangle geometry. Some famous names, and some not so famous: Euclid, Apollonius, Heron, Pythagoras, Menelaus, Pascal, Pierre de Fermat, Ceva, Steiner, Feuerbach, Soddy, Coxeter, and even Napolean!
5. Paul Yiu, "A Tour of Triangle Geometry"
6. Wikipedia, "Triangle"