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Golden Ratio and Arithmetic Mean

Golden Ratio and Arithmetic Mean
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2019-08-10 01:40
Complex NumberGeometric Construction
This example demonstrates the relationship between the golden ratio and arithmetic mean. Key mathematical concepts include arithmetic mean and the golden ratio. Using commands like Point, Semicircle, Midpoint, etc., from the construction summary, relevant geometric shapes can be constructed. Suitable for students at the junior high school level and above to learn geometric construction and related concepts.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(xAxis) → C
  4. Point(xAxis) → D
  5. Point(xAxis) → E
  6. Point(xAxis) → F
  7. Point(xAxis) → G
  8. Point(xAxis) → H
  9. Point(xAxis) → I
  10. Point(xAxis) → J
  11. Semicircle(A, J) → c
  12. Midpoint(A, J) → K
  13. OrthogonalLine(K, xAxis) → f
  14. Mirror(H, J) → H'
  15. Mirror(J, H') → J'
  16. Mirror(H', J') → H''
  17. Mirror(J', H'') → J''
  18. Semicircle(B, J'') → d
  19. Midpoint(B, J'') → L
  20. OrthogonalLine(L, xAxis) → g
  21. Intersect(d, g, 1) → M
  22. Segment(L, M) → h
  23. Point(xAxis) → N
  24. Point(xAxis) → O
  25. Point(xAxis) → P
  26. Point(xAxis) → Q
  27. Point(xAxis) → R
  28. Point(xAxis) → S
  29. Semicircle(N, T) → e
  30. Midpoint(N, T) → U
  31. OrthogonalLine(U, xAxis) → i
  32. Intersect(e, i, 1) → V
  33. Segment(U, V) → j
  34. OrthogonalLine(Q, xAxis) → k
  35. Intersect(e, k, 1) → W
  36. Segment(Q, W) → l

AI Drawing Prompt Reference

Place several points along the x-axis. Draw a semicircle with the two outermost points as endpoints, and construct a perpendicular line through its midpoint. Reflect a middle point repeatedly about the semicircle's endpoints to obtain new points. Using one of these new points and another original point as endpoints, draw a second semicircle, construct its perpendicular bisector, and intersect it with the semicircle to get a point; connect this point to the semicircle's midpoint. Then draw another semicircle elsewhere, construct its perpendicular bisector to intersect it, and connect the midpoint to that intersection. Also, from another point on this semicircle, draw a perpendicular to the x-axis, intersect it with the semicircle, and connect the two points. This construction illustrates the relationship between the golden ratio and the arithmetic mean.
The prompt above was compiled from the original drawing commands and is for reference only.