Triangle Centroid


The geometric centroid (center of mass) of the polygon vertices of a triangle is the point G (sometimes also denoted M) 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


and homogeneous barycentric coordinates (1,1,1). It is Kimberling center X_2.

The centroid satisfies


The centroid of the triangle with trilinear vertices p_i:q_i:r_i for i=1, 2, 3 is given by


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

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

triangleKimberlingtriangle centroid
anticomplementary triangleX_2triangle centroid
circumnormal triangleX_3circumcenter
circumtangential triangleX_3circumcenter
contact triangleX_(354)Weill point
Euler triangleX_(381)midpoint of X_2 and X_4
excentral triangleX_(165)centroid of the excentral triangle
extouch triangleX_(210)X_(10)-Ceva conjugate of X_(37)
first Brocard triangleX_2triangle centroid
first Morley triangleX_(356)first Morley center
first Neuberg triangleX_2triangle centroid
incentral triangleX_(1962)bicentric sum of pu(32)
inner Napoleon triangleX_2triangle centroid
inner Vecten triangleX_2triangle centroid
medial triangleX_2triangle centroid
orthic triangleX_(51)centroid of orthic triangle
outer Napoleon triangleX_2triangle centroid
outer Vecten triangleX_2triangle centroid
reference triangleX_2triangle centroid
second Neuberg triangleX_2triangle centroid
Stammler triangleX_3circumcenter
tangential triangleX_(154)X_3-Ceva conjugate of X_6

If the sides of a triangle DeltaA_1A_2A_3 are divided by points P_1, P_2, and P_3 so that


then the centroid G_P of the triangle DeltaP_1P_2P_3 is simply G_A, the centroid of the original triangle DeltaA_1A_2A_3 (Johnson 1929, p. 250).


One Brocard line, triangle median, and symmedian (out of the three of each) are concurrent, with AOmega, CK, and BG meeting at a point, where Omega is the first brocard point and K is the symmedian point. Similarly, AOmega^', BG, and CK, where Omega^' 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 X. The triangles BXC, CXA, and AXB have equal areas iff X 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 ABC,


where d, d_A, d_B, and d_C 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 s_i^(-1),


where Delta is the area of the triangle. Let P be an arbitrary point, the polygon vertices be A_1, A_2, and A_3, and the centroid G. Then


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


The distances from various named centers include


where I is the incenter, H is the orthocenter, O is the circumcenter, K is the symmedian point, L is the de Longchamps point, N is the nine-point center, Na is the Nagel point, and Sp 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 Ge, triangle centroid G, and mittenpunkt M are collinear, with GeG:GM=2:1.


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

See also

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

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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.", C. "Encyclopedia of Triangle Centers: X(2)=Centroid.", 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.

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