In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. ( April 2013) ( Learn how and when to remove this template message) Please help to improve this article by introducing more precise citations. It is referred to as the 'center of mass' or 'balance point' of the triangle. C.This article includes a list of general references, but it lacks sufficient corresponding inline citations. To locate the centroid, draw each of the three medians (which connect the vertices of the triangle to the midpoints of the opposite sides). Stein, S.: Archimedes: What Did He Do Besides Cry Eureka? MAA, Washington, D. Sommerville, D.M.Y.: An Introduction to the Geometry of N Dimensions. Prasolov, V.V., Tikhomirov, V.M.: Geometry. Peterson, M.A.: The geometry of Piero della Francesca. Ostermann, A., Wanner, G.: Geometry by Its History. Martini, H., Weissbach, B.: Napoleon’s theorem with weights in n-space. Liberti, L., Lavor, C., Maculan, N., Mucherino, A.: Euclidean distance geometry and applications. Lawes, C.P.: Proof without words: the length of a triangle median via the parallelogram law. Krantz, S.G., McCarthy, J.E., Parks, H.R.: Geometric characterizations of centroids of simplices. Johnson, R.A.: Advanced Euclidean Geometry. Izumi, S.: Sufficiency of simplex inequalities. Hungerbühler, N.: Proofs without words: the area of the triangle of the medians has three-fourths the area of the original triangle. Honsberger, R.: Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Hersh, R.: Heron’s formula: what about a tetrahedron? Coll. Heath, T.L.: The Thirteen Books of Euclid’s Elements, 2nd edn. Hajja, M., Walker, P.: The Gergonne and Nagel centers of a tetrahedron. Hajja, M., Martini, H., Spirova, M.: New extensions of Napoleon’s theorem to higher dimensions. Hajja, M.: The Gergonne and Nagel centers of an n-dimensional simplex. Gerber, L.: The orthocentric simplex as an extreme simplex. solution, ibid, 86, 387 (2013)įiedler, M.: Isodynamic systems in Euclidean spaces and an n-dimensional analogue of a theorem by Pompeiu. 76, 193–203 (2003)Įdmonds, A.L., Hajja, M., Martini, H.: Coincidences of simplex centers and related facial structures. 68, 914–917 (1961)Ĭrabb, R.A.: Gaspard Monge and the Monge point of the tetrahedron. Springer, Berlin (1994)īlumenthal, L.M.: A budget of curiosa metrica. XYZ Press, LLC (2016)īalk, M.B., Boltyanskij, V.G.: Geometry of Masses. The theorem basically says that: The length of the centroid to the midpoint of the opposite side is 2 times the length of the centroid to the vertex. (1964)Īndreescu, T., Korsky, S., Pohoata, C.: Lemmas in Olympiad Geometry. If you connect a line from the midpoint of one side to the vertex opposite to that side (which is a median), then the centroid is where all 3 medians intersect. 125, 612–622 (2018)Īl-Afifi, G., Hajja, M., Hamdan, A., Krasopoulos, P.T.: Pompeiu-like theorems for the medians of a simplex. Springer, Berlin (2004)Īl-Afifi, G., Hajja, M., Hamdan, A.: Another n-dimensional generalization of Pompeiu’s theorem. Abrosimov, N.V., Makai Jr., E., Mednykh, A.D., Nikonorov, Y.G., Rote, G.: The infimum of the volumes of convex polytopes of any given facet areas is 0.
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