# 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.

### Barycenter and Barycentric Coordinates

- 3D Quadrilateral - a Coffin Problem
- Barycentric Coordinates
- Barycentric Coordinates: a Tool
- Barycentric Coordinates and Geometric Probability
- Ceva's Theorem
- Determinants, Area, and Barycentric Coordinates
- Maxwell Theorem via the Center of Gravity
- Bimedians in a Quadrilateral
- Simultaneous Generalization of the Theorems of Ceva and Menelaus
- Three glasses puzzle
- Van Obel Theorem and Barycentric Coordinates
- 1961 IMO, Problem 4. An exercise in barycentric coordinates
- Centroids in Polygon
- Center of Gravity and Motion of Material Points
- Isotomic Reciprocity
- An Affine Property of Barycenter
- Problem in Direct Similarity
- Circles in Barycentric Coordinates
- Barycenter of Cevian Triangle
- Concurrent Chords in a Circle, Equally Inclined

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Copyright © 1996-2018 Alexander Bogomolny64639348 |