Theorems on regular polygons

Zoltán Kovács, Feb 27, 2018

[url=http://www.jstor.org/stable/2324301?seq=1#page_scan_tab_contents]Watkins and Zeitlin (1993)[/url] published a method to simply compute the minimal polynomial of cos(2π/[i]n[/i]), based on the [url=https://en.wikipedia.org/wiki/Chebyshev_polynomials#First_kind]Chebyshev polynomials of the first kind[/url]. By using their method GeoGebra is able to prove various properties of regular [i]n[/i]-gons since November 2017. This collection demonstrates how useful this new feature is. As a further improvement of the Watkins-Zeitlin method, another software tool [url=ttps://github.com/kovzol/RegularNGons]RegularNGons[/url] can be used to automatically find interesting features of a regular polygon.


  • 1. A property of the regular heptagon
  • 2. Some properties of a regular heptagon
  • 3. Some properties of a regular nonagon
  • 4. A property of the regular nonagon
  • 5. Some interesting properties of the regular decagon
  • 6. Regular pentagon erected internally on a side of a regular decagon
  • 7. Touching pentagons
  • 8. Some properties of a regular 11-gon
  • 9. Some properties of a regular 13-gon
  • 10. Some properties of a regular 17-gon
  • 11. Unit lengths in the regular 19-gon
  • 12. A theorem on regular n-gons (n is even)

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A property of the regular heptagon

Consider the construction above. It is a regular 7-gon with some diagonals, their intersection points and a segment joining them. Remark that [i]HI[/i]=2[i]AB[/i], moreover [i]A[/i] is the midpoint of [i]HI[/i].[br]This theorem has been obtained automatically by using the software tool [url=https://github.com/kovzol/RegularNGons]RegularNGons[/url].
– Zoltán Kovács

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