# Centroids in Polygon: What is it about?

A Mathematical Droodle

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(The applet allows one to toy with N-gons. To change N, click a little off its vertical center line - left or right.)

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Copyright © 1996-2018 Alexander Bogomolny## Centroids in Polygon

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Let P = P_{1}P_{2} ... P_{N} be any N-gon. Imagine placing unit weights (material points) at its vertices. Assume Q_{i} is the *centroid* (barycenter, center of gravity) of the (N-1)-gon obtained from P by dropping vertex P_{i}, i = 1, 2, ..., N. Then the N lines P_{i}Q_{i} all meet in a point, say G, such that, for any i, _{i}G/GQ_{i} = N-1.

Note that, for N = 3, we get a familar case of the medians in a triangle divided in the ratio 2:1 by the triangle's centroid.

We may conclude that the N-gon Q = Q_{1}Q_{2} ... Q_{N} is similar to P and homothetic to it at G with the coefficient 1/(1-N).

Other geometric facts could be derived with the help of the (mechanical) idea of the center of gravity. See for example, the Paraxegon construction and its Paragon generalization and the page on Medians in a Quadrilateral.

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Copyright © 1996-2018 Alexander Bogomolny