The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville.

As of 6 February 2024, the list identifies over 61,000 triangle centers.[1]

Each point in the list is identified by an index number of the form X(n)—for example, X(1) is the incenter. The information recorded about each point includes its trilinear and barycentric coordinates and its relation to lines joining other identified points. Links to The Geometer's Sketchpad diagrams are provided for key points. The Encyclopedia also includes a glossary of terms and definitions.

Each point in the list is assigned a unique name. In cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, the 770th point in the list is named point Acamar.

Notable points

The first 10 points listed in the Encyclopedia are:

ETC reference Name Definition
X(1) Incenter center of the incircle
X(2) Centroid intersection of the three medians
X(3) Circumcenter center of the circumscribed circle
X(4) orthocenter intersection of the three altitudes
X(5) nine-point center center of the nine-point circle
X(6) symmedian point intersection of the three symmedians
X(7) Gergonne point symmedian point of contact triangle
X(8) Nagel point intersection of lines from each vertex to the corresponding semiperimeter point
X(9) Mittenpunkt symmedian point of the triangle formed by the centers of the three excircles
X(10) Spieker center center of the Spieker circle

Other points with entries in the Encyclopedia include:

ETC reference Name
X(11) Feuerbach point
X(13) Fermat point
X(15), X(16) first and second isodynamic points
X(17), X(18) first and second Napoleon points
X(19) Clawson point
X(20) de Longchamps point
X(21) Schiffler point
X(22) Exeter point
X(39) Brocard midpoint
X(40) Bevan point
X(175) Isoperimetric point
X(176) Equal detour point

Similar, albeit shorter, lists exist for quadri-figures (quadrilaterals and systems of four lines) and polygon geometry.

See also


  1. ^ Kimberling, Clark. "Part 31: Centers X(52001) - X(54000)". Encyclopedia of Triangle Centers. Retrieved February 6, 2024.