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Golden Rectangle - Construction

Golden Rectangle - Construction
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2017-01-07 08:38
PolygonAngleCoordinate GeometryTransformationGeometric Construction
This example demonstrates the step-by-step geometric construction of a golden rectangle, where the ratio of its sides equals the golden ratio (approximately 1.618). It visually illustrates the relationship between the golden ratio, similar triangles, and rectangle properties. The construction uses GeoGebra commands such as Polygon, Circle, Ray, Intersect, Segment, OrthogonalLine, Midpoint, and Angle to build a square, its diagonal, perpendicular lines, and intersections, ultimately forming the golden rectangle with auxiliary lines. This is suitable for high school or early college students learning about the golden ratio in geometry, and serves as a classic construction exercise for classroom demonstration or independent exploration.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Circle(E, C) → c
  3. Ray(A, B) → j
  4. Intersect(c, j, 1) → F
  5. Segment(E, C) → k
  6. OrthogonalLine(F, j) → l
  7. OrthogonalLine(C, g) → m
  8. Intersect(l, m) → G
  9. Midpoint(k) → H
  10. Segment(E, B) → n
  11. Segment(A, B) → p
  12. Segment(A, E) → q
  13. Angle(B, F, G) → α
  14. Segment(I, J) → r
  15. Midpoint(r) → K
  16. Segment(L, M) → s
  17. Segment(O, N) → t
  18. Segment(A, F) → b
  19. Segment(F, G) → d
  20. Segment(G, D) → e
  21. Segment(Q, P) → f_1
  22. Segment(T, U) → g_1
  23. Segment(R, S) → h_1
  24. Midpoint(f_1) → V

AI Drawing Prompt Reference

Draw a square, then draw a circle centered at the midpoint of one side. Extend one side of the square as a ray until it intersects the circle. From the intersection point, draw a perpendicular line to the ray. Also, draw a perpendicular line from one corner of the square to the side. The intersection of these two perpendicular lines gives another point. Connect the center of the circle to this intersection point and find the midpoint of this segment. Finally, connect various points and lines to form a golden rectangle, highlighting the relationships and auxiliary constructions involved.
The prompt above was compiled from the original drawing commands and is for reference only.