The centroid is the point of concurrency of the three medians of a triangle. A median is the line segment from a vertex of a triangle to the midpoint of the opposite side.
The centroid is also called the center of mass. If a triangle is made of a metallic plate of uniform density and thickness, it would be able to balance on one point; that point is the centroid. Any line drawn through the centroid bisects the triangle's area in half.
Some geometers refer to it as the geocenter. It is a rockstar of triangle centers, like the incenter, circumcenter, excenters, and orthocenter, because it's given its own letter, G. In Kimberling triangle center notation, it is X2. (X1 is the incenter.)
Cartesian Coordinates: The centroid's coordinates are G = (GX, GY), where
GX = (AX + BX + CX)/2 (1a)
GY = (AY + BY + CY)/2 (1b)
and A, B, and C are the coordinates of the reference triangle's vertices.
Barycentric Coordinates: 1:1:1
Trilinear Coordinates: bc:ca:ab
Euler Line: The centroid, the circumcenter, and the orthocenter are all collinear, somewhat amazingly, since they are constructed in entirely different ways. That they are collinear, and furthermore that the distance between the centroid and the orthocenter is always twice the distance between the centroid and the circumcenter, was proved by the great Swiss mathematician, Leonhard Euler. The line segment HO' is named the Euler line in his honor.
References: Useful books and references on geometry
- H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
- Dan Pedoe, Geometry: A Comprehensive Course
- J.L. Heilbron, Geometry Civilized, ©2000
- David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991
- Melvin Hausner, A Vector Space Approach to Geometry, ©1965
- Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996
Ch. 4, Geometry,
esp. § 4.5.1, “Triangles,” p. 271
- Wikipedia, "Centroid"
- Wikipedia, "Mass Point Geometry" The article version, dated Jan 21, 2012, makes the ridiculous claim that "Though modern mass point geometry was developed in the 1960s by New York high school students,4 the concept has been found to have been used as early as 1827 by August Ferdinand Möbius in his theory of homogenous coordinates.", and gives as reference a textbook: Rhoad, R., Milauskas, G., and Whipple, R. Geometry for Enjoyment and Challenge, McDougal, Littell & Company, (c)1991
- Math Open Reference, "Centroid of a Triangle"
- Math Open Reference, "The Euler Line"
- Weisstein, Eric W. "Triangle Centroid" From MathWorld--A Wolfram Web Resource.
- Clark Kimberling, "Centroid" University of Evansville, Evansville, IN
- Alexander Bogomolny, "A Characteristic Property of Centroid," Cut the Knot
- Antonio Guiterrez, "Ceva's Theorem"
- P. Ballew, "Centroid"