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

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Copyright © 1996-2015 Alexander Bogomolny

Centroids in Polygon

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Let P = P₁P₂ ... 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_iQ_i all meet in a point, say G, such that, for any i, P_iG/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 = Q₁Q₂ ... 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.

Barycenter and Barycentric Coordinates

1. 3D Quadrilateral - a Coffin Problem
2. Barycentric Coordinates
3. Barycentric Coordinates: a Tool
4. Barycentric Coordinates and Geometric Probability
   • Stick Broken Into Three Pieces (Trilinear Coordinates)
   • Stick Broken Into Three Pieces (Cartesian Coordinates)
5. Ceva's Theorem
6. Determinants, Area, and Barycentric Coordinates
7. Maxwell Theorem via the Center of Gravity
8. Bimedians in a Quadrilateral
9. Simultaneous Generalization of the Theorems of Ceva and Menelaus
   • Theorems of Ceva and Menelaus, an Illustrated Generalization
10. Three glasses puzzle
11. Van Obel Theorem and Barycentric Coordinates
12. 1961 IMO, Problem 4. An exercise in barycentric coordinates
13. Centroids in Polygon
14. Center of Gravity and Motion of Material Points
15. Area of Cevian Triangle

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Copyright © 1996-2015 Alexander Bogomolny