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Square + Rhombus = Constant Surprise! (GoGeometry Action 71)

Square + Rhombus = Constant Surprise! (GoGeometry Action 71)
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2017-06-29 01:19
PolygonAngleTransformationGeometric ConstructionSlider Animation
① This dynamic GeoGebra example illustrates a surprising constant relationship that exists when a square is built on one side of any rhombus. ② Core mathematical concepts include rhombus properties, square construction, rotation, similarity, angle relationships, and geometric invariants. ③ Construction highlights: using GeoGebra commands such as Rotate, Ray, Circle, Intersect, Line, Segment, Angle, Dilate, Polygon, CircleSector, OrthogonalLine, Min, Vector, Translate, and Point, the applet builds a rhombus, a square on its side, and auxiliary elements, allowing dynamic manipulation to reveal the constant. ④ Suitable for middle/high school geometry classes, math exploration, or teacher demonstrations to foster understanding of geometric transformations and invariants.

GeoGebra Commands & Parameters

  1. Rotate(B, (a * π / 180), A) → B'
  2. Ray(A, B') → j
  3. Circle(A, B) → c
  4. Intersect(c, j) → E,
  5. Line(B, E) → m
  6. Segment(A, B) → f
  7. Circle(B, f) → h
  8. Circle(E, f) → d
  9. Intersect(d, h, 1) → C
  10. Line(A, C) → l
  11. Segment(E, C) → i
  12. Circle(B, f) → e
  13. Angle(B, A, B') → α
  14. Segment(A, E) → g
  15. Segment(B, C) → k
  16. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], B) → A'
  17. Segment(A, A') → j_1
  18. Rotate(A, If[1 < u ≤ 2, ((u - 1) * α), α], A) → A'_1
  19. Rotate(A', If[1 < u ≤ 2, ((u - 1) * α), α], A) → A''
  20. Segment(A'_1, A'') → k_1
  21. Rotate(A'_1, If[2 < u ≤ 3, ((u - 2) * (π - α)), π - α], A) → Q
  22. Rotate(A'_1, If[2 < u ≤ 3, ((u - 2) * (π - α)), π - α], E) → R
  23. Rotate(A'', If[2 < u ≤ 3, ((u - 2) * (π - α)), π - α], E) → A'''_1
  24. Segment(R, A'''_1) → m_1
  25. Rotate(R, If[3 < u ≤ 4, ((u - 3) * α), α], C) → R'
  26. Rotate(A'''_1, If[3 < u ≤ 4, ((u - 3) * α), α], C) → S
  27. Segment(R', S) → l_1
  28. Polygon(A, B, C, E) → poly1, r_1, s_1, t_1, a_1
  29. Circle(A, (0.15 * f)) → d_1
  30. Segment(A, A'') → q_1
  31. Intersect(d_1, q_1) → T,
  32. Intersect(d_1, f) → U,
  33. CircleSector(A, U, T) → e_1
  34. Segment(A, A'') → r_2
  35. Segment(A'', A) → b_2
  36. Segment(A, B) → f_3
  37. Segment(R, B) → h_3
  38. OrthogonalLine(B, b_2) → j_3
  39. Intersect(b_2, j_3) → A_1
  40. Line(A, R) → r
  41. Line(B, A'') → t
  42. OrthogonalLine(R, m_1) → k_3
  43. Segment(A, B) → m_3
  44. Circle(R, m_3) → c_3
  45. Circle(B, m_3) → d_3
  46. OrthogonalLine(B, l_1) → n_3
  47. Intersect(d_3, n_3, 2) → D_1
  48. OrthogonalLine(D_1, n_3) → p_3
  49. OrthogonalLine(R, l_1) → q_3
  50. Intersect(p_3, q_3) → E_1
  51. Segment(B, D_1) → r_3
  52. Angle(E_1, D_1, B) → ε
  53. Dilate(B, If[4 < u ≤ 5, 5 - u, 0], D_1) → B'_1
  54. Segment(B, B'_1) → s_3
  55. Rotate(B, If[5 < u ≤ 6, ((-(u - 5)) * π / 2), ((-π)) / 2], D_1) → B'_2
  56. Rotate(B'_1, If[5 < u ≤ 6, ((-(u - 5)) * π / 2), ((-π)) / 2], D_1) → F_1
  57. Segment(B'_2, F_1) → t_3
  58. Rotate(B'_2, If[6 < u ≤ 7, ((-(u - 6)) * π / 2), ((-π)) / 2], E_1) → G_1
  59. Rotate(E_1, If[6 < u ≤ 7, ((-(u - 6)) * π / 2), ((-π)) / 2], E_1) → H_1
  60. Rotate(B'_2, If[6 < u ≤ 7, ((-(u - 6)) * π / 2), ((-π)) / 2], E_1) → I_1
  61. Rotate(F_1, If[6 < u ≤ 7, ((-(u - 6)) * π / 2), ((-π)) / 2], E_1) → J_1
  62. Segment(I_1, J_1) → a_3
  63. Angle(R, E_1, D_1) → ζ
  64. Polygon(B, D_1, E_1, R) → poly2, b_3, d_4, e_3, r_4
  65. Line(D_1, A) → p
  66. Line(A'', B) → f_1
  67. Intersect(p, f_1) → F
  68. Segment(F, D_1) → h_1
  69. Segment(F, A) → i_1
  70. Segment(F, B) → n_1
  71. Min({h_1, i_1, n_1}) → o
  72. Circle(R, (0.4 * o)) → c_1
  73. Line(R, B) → b_1
  74. Line(R, E_1) → f_2
  75. Intersect(c_1, b_1) → H, I
  76. Intersect(c_1, f_2) → J, K
  77. Rotate(B, 45°, R) → B'_3
  78. Angle(B, R, B'_3) → γ
  79. Ray(R, B'_3) → g_2
  80. Intersect(c_1, g_2) → L, D
  81. Dilate(A'', If[7 < u ≤ 8, 8 - u, 0], B) → A'''
  82. Segment(A'', A''') → g_1
  83. Dilate(B, If[8 < u ≤ 9, 9 - u, 0], F) → B'_4
  84. Segment(B, B'_4) → h_2
  85. Dilate(D_1, If[8 < u ≤ 9, 9 - u, 0], F) → G
  86. Segment(D_1, G) → i_2
  87. Dilate(F, If[9 < u ≤ 10, 10 - u, 0], A) → F'
  88. Segment(F, F') → j_2
  89. Dilate(R, If[9 < u ≤ 10, 10 - u, 0], A) → R'_1
  90. Dilate(R, If[10 < u ≤ 11, 11 - u, 0], D_1) → R'_2
  91. Segment(R, R'_2) → k_2
  92. Rotate(K, If[11 < u ≤ 12, ((-(u - 11)) * π / 4), ((-π)) / 4], R) → K'
  93. CircleSector(R, K', K) → c_2
  94. Rotate(c_2, If[13 < u ≤ 14, ((-(u - 13)) * π / 4), ((-π)) / 4], R) → d_2
  95. Dilate(R, If[14.5 < u ≤ 15.5, 15.5 - u, 0], F) → R'_3
  96. Vector(R, R'_3) → v
  97. Translate(d_2, v) → e_2
  98. Line(F, k_2) → l_2
  99. Point(l_2) → M
  100. Angle(A, F, M) → β
  101. Rotate(e_2, If[15.5 < u ≤ 16.5, ((-(u - 15.5)) * β), (-β)], F) → p_2

AI Drawing Prompt Reference

Draw a rhombus with one side as the base, and construct a square outward on that side. At one vertex of the rhombus, create a ray by rotating a point, and intersect it with a circle centered at that vertex to determine another vertex, completing the rhombus. Then, build a dynamically rotating square on the rhombus's side, with rotation controlled by a slider, progressively revealing intersection points, perpendicular lines, auxiliary circles, and circular sectors as the square rotates. The entire figure demonstrates a constant geometric relationship between the square and the rhombus through animation.
The prompt above was compiled from the original drawing commands and is for reference only.