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Generalized Mean and Trapezoid: Visualizing Inequalities

Generalized Mean and Trapezoid: Visualizing Inequalities
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2016-02-14 04:34
PolygonFunctionInequalityGeometric ConstructionSlider Animation
This GeoGebra example dynamically visualizes the inequalities among generalized means (such as arithmetic, geometric, and harmonic means) within a trapezoid. It demonstrates how different power means relate to the geometry of a trapezoid, incorporating conic sections (e.g., hyperbolas) and the unit circle to represent mean values. The construction uses commands like RotateText, Line, Point, Polygon, If, Intersect, Midpoint, and Segment to build the trapezoid ABCD, its diagonals, midpoints, and parallel auxiliary lines. Multiple mean points (k_H, k_G, k_Q) are defined and linked to sliders, allowing users to explore how the means change with parameters. This resource is suitable for high school and undergraduate students studying inequalities, mean value theorems, and the interplay between algebra and geometry. It serves as an interactive tool for classroom demonstrations or independent mathematical exploration.

GeoGebra Commands & Parameters

  1. RotateText("uStas\;\&\;Т.С.", 90*°) → надпись01
  2. Line(K + (0, 8), Line[K, L]) → cd
  3. Point(cd) → M
  4. Polygon(K, L, M, N) → s1, KL, LM, MN, NK
  5. If(NM ≠ 0, 1 / qq, 0) → k_G
  6. Intersect(Line[K, N], Line[L, M]) → O
  7. Midpoint(K, N) → A
  8. Midpoint(L, M) → B
  9. Segment(A, B) → a
  10. If(NM ≠ 9, Dilate[N, k_H, O], A) → N'_2
  11. If(NM ≠ 9, Dilate[M, k_H, O], B) → M'_2
  12. Segment(N'_2, M'_2) → h
  13. If(NM ≠ 9, Dilate[N, k_G, O], A) → N'
  14. If(NM ≠ 9, Dilate[M, k_G, O], B) → M'
  15. Segment(N', M') → g
  16. If((NM ≠ 0) ∧ (NM ≠ 9), Dilate[N, k_Q, O], If[NM ≟ 0, Dilate[K, k_Q, O], A]) → N'_1
  17. If((NM ≠ 0) ∧ (NM ≠ 9), Dilate[M, k_Q, O], If[NM ≟ 0, Dilate[L, k_Q, O], B]) → M'_1
  18. Segment(N'_1, M'_1) → q
  19. Segment(K, M) → e
  20. Segment(L, N) → f
  21. Intersect(e, f) → H
  22. Polygon(K, L, M'_1, N'_1) → s2, k, l, m'_1, n'_1
  23. Polygon(N'_1, M'_1, M, N) → s3, n'_2, m'_2, m, n