# 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.)

Explanation

## Centroids in Polygon

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

Let P = P1P2 ... PN be any N-gon. Imagine placing unit weights (material points) at its vertices. Assume Qi is the centroid (barycenter, center of gravity) of the (N-1)-gon obtained from P by dropping vertex Pi, i = 1, 2, ..., N. Then the N lines PiQi all meet in a point, say G, such that, for any i, PiG/GQi = 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 = Q1Q2 ... QN 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.