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SolveMyMaths Action 2! Dynamic Geometry Optimization

SolveMyMaths Action 2! Dynamic Geometry Optimization
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2018-06-08 08:14
CirclePolygonOptimizationGeometric ConstructionSlider Animation
① This interactive GeoGebra example dynamically illustrates a geometry optimization problem posed by Ed Southall (UK) on Twitter, featuring a square with inscribed circles, tangents, and adjustable proportions. ② Core mathematical concepts include squares, tangent lines, roots, optimization problems, ratios, and boxplots for data distribution, blending geometric construction with extremum seeking. ③ Construction highlights: using GeoGebra commands such as Polygon, Line, Intersect, Segment, Dilate, OrthogonalLine, Distance, Circle, Rotate, Angle, Ray, CircleSector, Midpoint, Vector, and Translate, the model builds polygons, lines, circles, angles, sectors, midpoints, and vectors, all controlled by sliders for dynamic manipulation. ④ Suitable for high school and above, this resource helps students and teachers explore geometric optimization, tangent properties, and ratio relationships, and serves as an interactive tool for math modeling or competition preparation.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 8) → poly1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
  2. Line(A, B) → n
  3. Line(D, C) → p
  4. Intersect(n, p) → I
  5. Polygon(D, I, 4) → poly2, q, r, s, t, J, K
  6. Segment(A, H) → a
  7. Segment(H, G) → b
  8. Segment(G, F) → c
  9. Segment(F, E) → d
  10. Segment(E, D) → e
  11. Segment(C, B) → f_1
  12. Segment(D, I) → g_1
  13. Segment(D, K) → h_1
  14. Segment(K, J) → i_1
  15. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], J) → C'_1
  16. Segment(C, C'_1) → j_1
  17. OrthogonalLine(H, n) → k_1
  18. Intersect(n, k_1) → L
  19. Dilate(H, If[1 < o ≤ 2, 2 - o, 0], L) → H'
  20. Segment(H, H') → l_1
  21. Line(H', n) → m_1
  22. Intersect(m_1, a) → M
  23. Polygon(H, H', M) → t1, m_2, h_2, h'
  24. Segment(H', M) → n_1
  25. Distance(L, A) → u
  26. Circle(A, (0.8 * u)) → c_1
  27. Intersect(c_1, n) → N, O
  28. Rotate(H, 22.5°, A) → H'_1
  29. Angle(H, A, H'_1) → α
  30. Ray(A, H'_1) → p_1
  31. Intersect(p_1, l_1) → P
  32. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], P) → A'
  33. Segment(A, A') → q_1
  34. Rotate(N, If[3 < o ≤ 4, ((-(o - 3)) * π / 8), ((-π)) / 8], A) → N'
  35. CircleSector(A, N', N) → d_1
  36. Midpoint(F, B) → Q
  37. Midpoint(D, J) → R
  38. Rotate(d_1, If[4.5 < o ≤ 5.5, ((-(o - 4.5)) * π / 8), ((-π)) / 8], A) → e_1
  39. Dilate(A, If[6 < o ≤ 7, 7 - o, 0], Q) → A'_1
  40. Vector(A, A'_1) → v
  41. Translate(e_1, v) → r_1
  42. Dilate(Q, If[7 < o ≤ 8, 8 - o, 0], J) → Q'
  43. Vector(Q, Q') → w
  44. Translate(r_1, w) → s_1
  45. Rotate(s_1, If[7 < o ≤ 8, (((-(o - 7)) * 15) * π / 8), π / 8], Q') → t_1

AI Drawing Prompt Reference

Draw a square and two tangent circles inside it, with the radius of one circle adjustable. First, draw a square, then choose a point on the diagonal as the center of a circle with an adjustable radius, making it tangent to two sides of the square. Next, adjust the center position to draw another circle that is tangent to the first circle and also tangent to two sides of the square. Finally, adjust the center position and radius to observe the changes in the relationship between the square and the circles.
The prompt above was compiled from the original drawing commands and is for reference only.