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Square and Equilateral Triangle

Square and Equilateral Triangle
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2019-03-21 00:46
PolygonTrigonometryGeometric Construction
In collections Cutting up squares
This example demonstrates the geometric relationship and construction between a square and an equilateral triangle. It involves core mathematical concepts such as polygons, geometric transformations, symmetry, and angles. The construction uses GeoGebra commands and objects like Intersect (intersection), Point, Polygon, Segment, Midpoint, Circle, Ray, OrthogonalLine, Rotate, and Line, as summarized. It is suitable for high school mathematics teaching on geometric construction and transformations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 3) → poly1, a, a_1, b, C
  4. Segment(A, B) → c
  5. Segment(B, C) → d
  6. Segment(C, A) → e
  7. Midpoint(d) → D
  8. Midpoint(e) → E
  9. Circle(D, C) → f
  10. Ray(A, D) → g
  11. Intersect(f, g) → F_1, F
  12. Midpoint(A, F) → G
  13. Circle(G, F) → h
  14. Ray(B, C) → i
  15. Intersect(h, i) → H,
  16. Circle(D, H) → k
  17. Intersect(k, c, 1) → J
  18. Circle(J, Segment[C, D]) → p
  19. Intersect(p, c, 1) → K
  20. Segment(J, D) → j
  21. OrthogonalLine(E, j) → l
  22. Intersect(l, j) → N
  23. OrthogonalLine(K, j) → m
  24. Intersect(m, j) → R
  25. Segment(E, N) → n
  26. Segment(K, R) → q
  27. Polygon(C, D, N, E) → poly2, c_1, d_1, n_1, e_1
  28. Polygon(D, B, K, R) → poly3, d_2, b_1, k_1, r
  29. Polygon(J, K, R) → poly4, r_1, j_1, k_2
  30. Polygon(A, J, N, E) → poly5, a_2, j_2, n_2, e_2
  31. Rotate(C, (-α), D) → C'_1
  32. Rotate(D, (-α), D) → D'
  33. Rotate(N, (-α), D) → N'_1
  34. Rotate(E, (-α), D) → E'_1
  35. Polygon(C'_1, D', N'_1, E'_1) → poly2', c', d', n', e'_1
  36. Rotate(J, α, K) → J'_1
  37. Rotate(K, α, K) → K'
  38. Rotate(R, α, K) → R'
  39. Polygon(J'_1, K', R') → poly4', r', j', k'
  40. Rotate(A, α, K') → A'
  41. Rotate(J, α, K') → J'
  42. Rotate(N, α, K') → N'
  43. Rotate(E, α, K') → E'
  44. Polygon(A', J', N', E') → poly5', a', j'_1, n'_1, e'
  45. Intersect(b, e) → I
  46. Intersect(a_1, d) → L
  47. Intersect(b, e) → M
  48. Intersect(a_1, d) → O
  49. Point(poly1) → P
  50. Line(N, R) → s
  51. Circle(D', N') → t
  52. Line(N, D') → f_1
  53. Circle(D', N) → g_1
  54. Intersect(g_1, f_1, 2) → Q
  55. Polygon(Q, R, 4) → poly6, q_1, r_2, h_1, i_1, S, T
  56. Circle(Q, D') → p_1
  57. Point(poly6) → U
  58. Polygon(D', Q, U, B) → poly7, d'_1, q_2, u, b_2
  59. Circle(K', J) → s_1
  60. Point(poly6) → V
  61. Polygon(K', V, S) → poly8, s_2, k'_1, v
  62. Polygon(B, U, T, V) → poly9, b_3, u_1, t_1, v_1
  63. Rotate(B, α - 360°, V) → B'
  64. Rotate(U, α - 360°, V) → U'
  65. Rotate(T, α - 360°, V) → T'
  66. Rotate(V, α - 360°, V) → V'
  67. Polygon(B', U', T', V') → poly9', b', u', t', v'