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(Mathematics - Affine Geometry)

Centroid of a triangle

The centroid of a triangle is the concurrent point of the three medians, or lines that pass through a vertex and the midpoint of the side opposite of that vertex. The centroid G of a triangle ABC is the point G = (1/3)(A + B + C), which is equal to G = (1/3)(A) + (2/3)(1/2)(B + C), which shows that G lies on the median that passes A and the midpoint A' of B and C, at a point in between A and A' that is closer to A' than it is to A by half the distance. Because of the symmetry of the equation, G lies on all three medians.

Centroid of a finite set of points

To further generalise the definition, for a finite set of points A1, A2, … An, the centroid is defined as (1/n)(A1 + A2 + … An). In fact, a midpoint is the centroid of two points. An interesting fact is that the centroid of the roots of a complex polynomial of degree greater than 1 is the same as the centroid of the roots of the derivative function.

Centroid of weighted points

If points A1, A2, … An had weights w1, w2, … wn, the centroid G of the set of weighted points is: G = (1/wtotal)(w1A1 + w2A2 + … wnAn) where wtotal = w1 + w2 + … wn.

We can partition the set of weighted points {A1, … An} into two sets, α and β. Suppose that sets A, α, β have total weights Aw, αw, βw and centroids AG, αG, βG. Then it turns out that AG = (1/Aw) ( αw * αG + βw * βG ) . G lies on the line that passes αG and βG. If we consider a triangle centroid to be the centroid of three points with weights 1, then by the above equation it is clear that the centroid lies on the medians at a location that leans closer towards the midpoints.

Other terms for centroid

Another term for the centroid of weighted points is affine combination.

When the weights of an affine combination are nonnegative, then the combination is called a convex combination.

Another term for the centroid of non-weighted points on the Real line is arithmetic mean.

In physics, centroid is also known as center of mass.

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

1. H.S.M. Coxeter, Introduction to Geometry, 2nd Ed., (c) 1969
2. Dan Pedoe, Geometry: A Comprehensive Course
3. J.L. Heilbron, Geometry Civilized, ©2000
4. David Wells, Ed., The Penguin Dictionary of Curious and Interesting Geometry, ©1991
5. Melvin Hausner, A Vector Space Approach to Geometry, ©1965
6. Daniel Zwillinger, Ed., The CRC Standard Mathematical Tables and Formulae, 30th Ed, ©1996
Ch. 4, Geometry,
esp. § 4.5.1, “Triangles,” p. 271

Internet References

1. Wikipedia, "Centroid"
2. 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
3. Math Open Reference, "Centroid of a Triangle"
4. Math Open Reference, "The Euler Line"
5. Weisstein, Eric W. "Triangle Centroid" From MathWorld--A Wolfram Web Resource.
6. Clark Kimberling, "Centroid" University of Evansville, Evansville, IN
7. Alexander Bogomolny, "A Characteristic Property of Centroid," Cut the Knot
8. Antonio Guiterrez, "Ceva's Theorem"
9. P. Ballew, "Centroid"

Cen"troid (?), n. [L. centrum + -oid.]

The center of mass, inertia, or gravity of a body or system of bodies.

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