The centroid (Center of Mass) of the Vertices of a Triangle is the point (or
) of intersection of the Triangle's three Medians, also called the Median Point
(Johnson 1929, p. 249). The centroid is always in the interior of the Triangle, and has Trilinear Coordinates

(1) |

(2) |

(3) |

Pick an interior point . The Triangles , , and have equal areas Iff
corresponds to the centroid. The centroid is located one third of the way from each 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 Median Point to the Vertices divide the whole into three equivalent
Triangles. In general, for any line in the plane of a Triangle ,

(4) |

(5) |

(6) |

(7) |

The centroid of the Perimeter of a Triangle is the triangle's Spieker Center (Johnson 1929, p. 249).

**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.

Johnson, R. A. *Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.*
Boston, MA: Houghton Mifflin, pp. 173-176 and 249, 1929.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' *Math. Mag.* **67**, 163-187, 1994.

Kimberling, C. ``Centroid.'' http://cedar.evansville.edu/~ck6/tcenters/class/centroid.html.

© 1996-9

1999-05-26