The Encyclopedia of Triangle Centers (ETC) is an on-line list of more than 3,000 points or "centers

Triangle center

In geometry a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Each of...

" associated with the geometry of a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

. It is maintained by Clark Kimberling

Clark Kimberling

Clark Kimberling is a mathematician, musician, and composer. He has been a mathematics professor since 1970 at the University of Evansville. His research interests include triangle centers, integer sequences, and hymnology.Kimberling received his Ph.D. in mathematics in 1970 from the Illinois...

, Professor of Mathematics at the University of Evansville

University of Evansville

The University of Evansville is a small, private university with approximately 3,050 students located in Evansville, Indiana. Founded in 1854 as Moores Hill College, it is located near the interchange of the Lloyd Expressway and U.S. Route 41. It is affiliated with the United Methodist Church...

.

Each point in the list is identified by an index number of the form X(n) — for example, X(1) is the incentre. The information recorded about each point includes its trilinear

Trilinear coordinates

In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative distances from the three sides of the triangle. Trilinear coordinates are an example of homogeneous coordinates...

 and barycentric coordinates

Barycentric coordinates (astronomy)

In astronomy, barycentric coordinates are non-rotating coordinates with origin at the center of mass of two or more bodies.The barycenter is the point between two objects where they balance each other. For example, it is the center of mass where two or more celestial bodies orbit each other...

 and its relation to lines joining other identified points. Links to Geometer's Sketchpad diagrams are provided for key points. The Encyclopedia also includes a glossary of terms and definitions.

The first 10 points listed in the Encyclopedia are:

ETC reference	Name	Definition
X(1)	incentre	centre of the incircle Incircle and excircles of a triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides...
X(2)	centroid Centroid In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...	intersection of the three medians Median (geometry) In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side...
X(3)	circumcentre	centre of the circumscribed circle Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter....
X(4)	orthocentre	intersection of the three altitudes Altitude (triangle) In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to a line containing the base . This line containing the opposite side is called the extended base of the altitude. The intersection between the extended base and the altitude is called the foot of the...
X(5)	nine-point centre	centre of the nine-point circle Nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant points defined from the triangle...
X(6)	symmedian point	intersection of the three symmedian Symmedian Symmedians are three particular geometrical lines associated with every triangle. They are constructed by taking a median of the triangle , and reflecting the line over the corresponding angle bisector... s
X(7)	Gergonne point	symmedian point of contact triangle
X(8)	Nagel point Nagel point In geometry, the Nagel point is a point associated with any triangle. Given a triangle ABC, let TA, TB, and TC be the extouch points in which the A-excircle meets line BC, the B-excircle meets line CA, and C-excircle meets line AB, respectively...	intersection of lines from each vertex to the corresponding semiperimeter Semiperimeter In geometry, the semiperimeter of a polygon is half its perimeter. Although it has such a simple derivation from the perimeter, the semiperimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name...  point
X(9)	Mittenpunkt Mittenpunkt In geometry, the mittenpunkt of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It is defined as the symmedian point of the excentral triangle of the given triangle. It is also the centroid of the Mandart...	symmedian point of the triangle formed by the centres of the three excircles Incircle and excircles of a triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides...
X(10)	Spieker centre	centre of the Spieker circle Spieker circle In geometry, the incircle of the medial triangle of a triangle ABC is the Spieker circle. Its center, the Spieker center, is the center of mass of the boundary of triangle ABC as well as being the incenter of the medial triangle....

Other points with entries in the Encyclopedia include:

ETC reference	Name
X(11)	Feuerbach point
X(13)	Fermat point Fermat point In geometry the Fermat point of a triangle, also called Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible...
X(15), X(16)	first and second isodynamic point Isodynamic point In Euclidean geometry, every triangle has two isodynamic points, usually denoted as S and S^'. These points are the common intersection points of the three circles of Apollonius associated with the triangle; hence, the line through these points is the common radical axis for these circles... s
X(20)	de Longchamps point De Longchamps point In geometry, the de Longchamps point of a triangle is the reflection of its orthocenter about its circumcenter. It is listed as X in the Encyclopedia of Triangle Centers. Its trilinear coordinates are\displaystyle\cos A - \cos B \cos C : \cos B - \cos C \cos A : \cos C - \cos A \cos BThe point is...
X(21)	Schiffler point Schiffler point In geometry, the Schiffler point of a triangle is a point defined from the triangle that is invariant under Euclidean transformations of the triangle. This point was first defined and investigated by Schiffler et al...
X(39)	Brocard midpoint Brocard point In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard , a French mathematician.-Definition:...

External links

• Encyclopedia of Triangle Centers
• Implementation of ETC points as Perl subroutines by Jason Cantarella