Equal Areas in Regular 2n-gons

The applet below illustrates the following theorem:

Let P be a point inside a regular 2n-gon. The lines joining P to the vertices of the polygon split the letter into 2n triangles. Then the sum of the areas of odd-numbered triangles equals that of the even-numbered triangles.


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?

Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Let P be a point inside a regular 2n-gon. The lines joining P to the vertices of the polygon split the letter into 2n triangles. Then the sum of the areas of odd-numbered triangles equals that of the even-numbered triangles.


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?

The claim is a consequence of a generalization of the Viviani's theorem that claims that in either equilateral or equiangular polygons the sum of distances from an arbitrary point in the plane of the polygon to its sidelines is constant.

Given a regular 2n-gon. Form two regular n-gons by extending the sides of the former. In each of the latter the distances from a point to the sides add up to a constant; in addition, the two constants are equal, and the sums in each of the n-gons add up to the sum of the altitudes of either odd-numbered or even-numbered triangles in the 2n-gon.

For the 4n-gon, one need not invoke the Viviani's theorem. As Matt Hendersen has observed, for the regular 4n-gons, the required result follows from the fact that the distance between any pair of opposite sides is exactly the same. (This is of course true for any regular 2n-gons, but for the 4n-gons the fact acquires extra significance. Do you see why?)


Related material
Read more...

  • Equilateral and 3-4-5 Triangles
  • Rusty Compass Construction of Equilateral Triangle
  • Equilateral Triangle on Parallel Lines
  • Equilateral Triangle on Parallel Lines II
  • When a Triangle is Equilateral?
  • Viviani's Theorem
  • Viviani's Theorem (PWW)
  • Tony Foster's Proof of Viviani's Theorem
  • Viviani in Isosceles Triangle
  • Viviani by Vectors
  • Slanted Viviani
  • Slanted Viviani, PWW
  • Morley's Miracle
  • Triangle Classification
  • Napoleon's Theorem
  • Sum of Squares in Equilateral Triangle
  • A Property of Equiangular Polygons
  • Fixed Point in Isosceles and Equilateral Triangles
  • Parallels through the Vertices of Equilateral Triangle
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    Copyright © 1996-2017 Alexander Bogomolny

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