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From Equiangular Hexagon to Ellipse

From Equiangular Hexagon to Ellipse
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2018-05-21 19:06
CirclePolygonOptimizationGeometric Construction
This example demonstrates an intriguing phenomenon where any equiangular hexagon can be transformed into an ellipse through specific geometric constructions. The core math concepts involved include properties of equiangular polygons, conic sections, geometric transformations, and optimization. The construction utilizes GeoGebra commands such as Ray, Rotate, Angle, Point, Intersect, Conic, Center, Segment, and Focus. It is suitable for high school and advanced learners exploring the intrinsic connection between geometric transformations and conic curves, as well as for math competitions or enrichment activities.

GeoGebra Commands & Parameters

  1. Ray(A, B) → f
  2. Rotate(A, (-(120°)), B) → A'
  3. Angle(A', B, A) → α
  4. Ray(B, A') → g
  5. Point(g) → C
  6. Rotate(B, (-(120°)), C) → B'
  7. Angle(B', C, B) → β
  8. Ray(C, B') → h
  9. Point(h) → D
  10. Rotate(C, (-(120°)), D) → C'
  11. Angle(C', D, C) → γ
  12. Ray(D, C') → i
  13. Point(i) → E
  14. Rotate(D, (-(120°)), E) → D'
  15. Angle(D', E, D) → δ
  16. Rotate(B, 120°, A) → B'_1
  17. Angle(B, A, B'_1) → ε
  18. Ray(A, B'_1) → j
  19. Ray(E, D') → k
  20. Intersect(j, k) → F
  21. Conic(A, B, C, D, E) → c
  22. Center(c) → G
  23. Segment(A, B) → l
  24. Segment(B, C) → m
  25. Segment(C, D) → n
  26. Segment(D, E) → p
  27. Segment(E, F) → q
  28. Segment(F, A) → r
  29. Focus(c) → H, I