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 © 1996-2017 Alexander Bogomolny

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.

Barycenter and Barycentric Coordinates

  1. 3D Quadrilateral - a Coffin Problem
  2. Barycentric Coordinates
  3. Barycentric Coordinates: a Tool
  4. Barycentric Coordinates and Geometric Probability
  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
  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. Isotomic Reciprocity
  16. An Affine Property of Barycenter
  17. Problem in Direct Similarity
  18. Circles in Barycentric Coordinates
  19. Barycenter of Cevian Triangle
  20. Concurrent Chords in a Circle, Equally Inclined

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Copyright © 1996-2017 Alexander Bogomolny

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