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Geometric Mean Demonstration by Tim Brzezinski

Geometric Mean Demonstration by Tim Brzezinski
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2017-03-04 07:01
CircleVectorGeometric Construction
In collections Congruence (CO) Cluster CCSS High School: Geometry (Congruence) Volume 2 Circles (Theorems) Triangle Theorems Constructions (Dynamic Illustrations)
This dynamic illustration demonstrates the definition and calculation of the geometric mean. It involves core mathematical concepts such as geometric mean, intersection, unit circle, vectors, bar chart, and cylinder. The construction utilizes GeoGebra commands like Ray, Point, Midpoint, Circle, Semicircle, OrthogonalLine, Intersect, Segment, Line, Angle, Rotate, CircleArc, Dilate, CircleSector, Vector, and Translate. It is suitable for high school mathematics teaching or self-learning to understand the geometric significance and applications of the geometric mean.

GeoGebra Commands & Parameters

  1. Ray(A, (1, 0)) → f
  2. Point(f) → B
  3. Ray(B, C) → g
  4. Point(g) → D
  5. Midpoint(A, D) → E
  6. Circle(E, D) → c
  7. Semicircle(A, D) → d
  8. OrthogonalLine(B, f) → h
  9. Intersect(d, h, 1) → F
  10. Segment(A, F) → i
  11. Line(D, F) → j
  12. Segment(F, D) → k
  13. Angle(A, F, D) → α
  14. Circle(F, 2) → e
  15. Intersect(e, j, 2) → G
  16. Angle(G, F, A) → β
  17. Circle(A, (0.15 * Min[i, k])) → p
  18. Circle(D, Radius[p]) → q
  19. Intersect(p, y = 0) → H, I
  20. Intersect(p, i, 1) → J
  21. Intersect(q, l, 1) → K
  22. Intersect(q, k, 1) → L
  23. Rotate(D, If[0 < a ≤ 1, (a * π), π], E) → D'
  24. CircleArc(E, D, D') → r
  25. Dilate(B, If[1 < a ≤ 2, 2 - a, 0], F) → F'
  26. Segment(B, F') → m
  27. Angle(F, B, A) → γ
  28. CircleSector(A, I, J) → s
  29. CircleSector(D, L, K) → t
  30. Dilate(A, If[3 < a ≤ 4, 4 - a, 0], F) → A'
  31. Dilate(D, If[3 < a ≤ 4, 4 - a, 0], F) → D'_1
  32. Segment(A, A') → n
  33. Segment(D, D'_1) → b
  34. Angle(B, A, F) → δ
  35. Angle(F, D, B) → ε
  36. Rotate(I, If[4 < a ≤ 5, ((a - 4) * δ), δ], A) → I'
  37. Rotate(K, If[4 < a ≤ 5, ((-(a - 4)) * ε), (-ε)], D) → K'
  38. CircleSector(D, K', K) → c_1
  39. CircleSector(A, I, I') → d_1
  40. Ray(F, G) → f_1
  41. Dilate(A, If[6 < a ≤ 7, 7 - a, 0], F) → A'_1
  42. Vector(A, A'_1) → u
  43. Translate(d_1, u) → e_1
  44. Rotate(e_1, If[7 < a ≤ 8, ((-(a - 7)) * π / 2), ((-π)) / 2], F) → g_1
  45. Dilate(D, If[8 < a ≤ 9, 9 - a, 0], F) → D'_2
  46. Vector(D, D'_2) → v
  47. Translate(c_1, v) → h_1
  48. Rotate(h_1, If[9 < a ≤ 10, ((a - 9) * π / 2), π / 2], F) → k_1
  49. Segment(A, B) → i_1
  50. Segment(B, D) → j_1

AI Drawing Prompt Reference

Create a dynamic illustration showing the definition and calculation of the geometric mean. Include a ray with two points on it, construct a circle and semicircle using these points and the ray's endpoint. Find the intersection of the semicircle with a line perpendicular to the ray, and connect this point to the ray's endpoint. Construct two sectors within the circle and use vectors and translations to create additional geometric objects, ultimately demonstrating the geometric significance of the geometric mean.
The prompt above was compiled from the original drawing commands and is for reference only.