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# Triangle Centroid

The geometric centroid (center of mass) of the polygon vertices of a triangle is the point (sometimes also denoted ) which is also the intersection of the triangle's three triangle medians (Johnson 1929, p. 249; Wells 1991, p. 150). The point is therefore sometimes called the median point. The centroid is always in the interior of the triangle. It has equivalent triangle center functions

 (1) (2) (3)

and homogeneous barycentric coordinates . It is Kimberling center .

The centroid satisfies

 (4)

The centroid of the triangle with trilinear vertices for , 2, 3 is given by

 (5)

(P. Moses, pers. comm., Sep. 7, 2005).

The following table summarizes the triangle centroids for named triangles that are Kimberling centers.

 triangle Kimberling triangle centroid anticomplementary triangle triangle centroid circumnormal triangle circumcenter circumtangential triangle circumcenter contact triangle Weill point Euler triangle midpoint of and excentral triangle centroid of the excentral triangle extouch triangle -Ceva conjugate of first Brocard triangle triangle centroid first Morley triangle first Morley center first Neuberg triangle triangle centroid incentral triangle bicentric sum of pu(32) inner Napoleon triangle triangle centroid inner Vecten triangle triangle centroid medial triangle triangle centroid orthic triangle centroid of orthic triangle outer Napoleon triangle triangle centroid outer Vecten triangle triangle centroid reference triangle triangle centroid second Neuberg triangle triangle centroid Stammler triangle circumcenter tangential triangle -Ceva conjugate of

If the sides of a triangle are divided by points , , and so that

 (6)

then the centroid of the triangle is simply , the centroid of the original triangle (Johnson 1929, p. 250).

One Brocard line, triangle median, and symmedian (out of the three of each) are concurrent, with , , and meeting at a point, where is the first brocard point and is the symmedian point. Similarly, , , and , where is the second Brocard point, meet at a point which is the isogonal conjugate of the first (Johnson 1929, pp. 268-269).

Pick an interior point . The triangles , , and have equal areas iff corresponds to the centroid. The centroid is located 2/3 of the way from each polygon vertex to the midpoint of the opposite side. Each median divides the triangle into two equal areas; all the medians together divide it into six equal parts, and the lines from the centroid to the polygon vertices divide the whole into three equivalent triangles. In general, for any line in the plane of a triangle ,

 (7)

where , , , and are the distances from the centroid and polygon vertices to the line.

A triangle will balance at the centroid, and along any line passing through the centroid. The trilinear polar of the centroid is called the Lemoine axis. The perpendiculars from the centroid are proportional to ,

 (8)

where is the area of the triangle. Let be an arbitrary point, the polygon vertices be , , and , and the centroid . Then

 (9)

If is the circumcenter of the triangle's centroid, then

 (10)

The distances from various named centers include

 (11) (12) (13) (14) (15) (16) (17) (18)

where is the incenter, is the orthocenter, is the circumcenter, is the symmedian point, is the de Longchamps point, is the nine-point center, is the Nagel point, and is the Spieker center.

The centroid lies on the Euler line and Nagel line. The centroid of the perimeter of a triangle is the triangle's Spieker center (Johnson 1929, p. 249). The symmedian point of a triangle is the centroid of its pedal triangle (Honsberger 1995, pp. 72-74).

The Gergonne point , triangle centroid , and mittenpunkt are collinear, with .

Given a triangle , construct circles through each pair of vertices which also pass through the triangle centroid . The triangle determined by the center of these circles then satisfies a number of interesting properties. The first is that the circumcircle and triangle centroid of are, respectively, the triangle centroid and symmedian point of the triangle (Honsberger 1995, p. 77). In addition, the triangle medians of and intersect in the midpoints of the sides of .

Circumcenter, Euler Line, Exmedian Point, Incenter, Nagel Line, Orthocenter

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## References

Carr, G. S. Formulas and Theorems in Pure Mathematics, 2nd ed. New York: Chelsea, p. 622, 1970.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 7, 1967.Dixon, R. Mathographics. New York: Dover, pp. 55-57, 1991.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., pp. 72-74 and 77, 1995.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 173-176, 249-250, and 268-269, 1929.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, 163-187, 1994.Kimberling, C. "Centroid." http://faculty.evansville.edu/ck6/tcenters/class/centroid.html.Kimberling, C. "Encyclopedia of Triangle Centers: X(2)=Centroid." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X2.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 62-63, 1893.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 150, 1991.

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Triangle Centroid

## Cite this as:

Weisstein, Eric W. "Triangle Centroid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TriangleCentroid.html