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三个平方和等于一个平方

三个平方和等于一个平方
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2019-08-18 19:02
测试改名0圆向量几何作图滑块动画
①本范例演示了三个平方和等于一个平方的几何现象；②涉及的核心数学概念包括勾股定理、平方、整数、对称性、除法、中值、直角三角形；③作图要点——从下方『构造摘要』提炼用到了Intersect（交点）、Point（点）、Polygon（多边形）、Midpoint（中点）、OrthogonalLine（垂线）、Circle（圆）、CircumcircleArc（外接圆弧）、Plane（平面）、Translate（平移）、Segment（线段）、OrthogonalPlane（正交平面）、Vector（向量）、Rotate（旋转）、Angle（角度）等GeoGebra命令/对象；④适合高中数学教学，用于理解和应用勾股定理及相关几何构造。

GeoGebra 命令及参数

  1. Intersect(zAxis, yAxis) → A
  2. Point(zAxis) → B
  3. Point(yAxis) → C
  4. Polygon(A, B, C) → t1, c, a, b
  5. Midpoint(b) → D
  6. OrthogonalLine(D, xOyPlane) → f
  7. Circle(f, A) → d
  8. Point(d) → E
  9. CircumcircleArc(A, E, C) → e
  10. Point(e) → F
  11. Polygon(A, F, C) → t2, c_1, a_1, f_1
  12. Polygon(A, F, 4, xOyPlane) → poly1, g, h, i, j, G, H
  13. Polygon(F, C, 4, xOyPlane) → poly2, k, l, m, n, I, J
  14. Plane(zAxis, yAxis) → p
  15. Polygon(A, B, 4, p) → poly3, q, r, s, t, K, L
  16. Polygon(B, C, 4, p) → poly4, g_1, h_1, i_1, j_1, M, N
  17. Polygon(C, A, 4, xOyPlane) → poly5, k_1, l_1, m_1, n_1, O, P
  18. Midpoint(H, F) → Q
  19. Midpoint(F, I) → R
  20. Midpoint(A, P) → S
  21. OrthogonalLine(R, p) → p_1
  22. Intersect(p_1, k) → T
  23. OrthogonalLine(R, xOyPlane) → q_1
  24. Plane(q_1, p_1) → r_1
  25. OrthogonalLine(R, r_1) → s_1
  26. Intersect(n, s_1) → U
  27. Intersect(p_1, m) → V
  28. Intersect(l, s_1) → W
  29. Translate(poly1, Vector[Q, If[0 < u ≤ 1, Dilate[Q, 1 - u, S], S]]) → poly1'
  30. Polygon(T, C, W, R) → q1, t_1, c_2, w, r_2
  31. Polygon(W, I, V, R) → q2, w_1, i_2, v, r_3
  32. Polygon(R, V, J, U) → q3, r_4, v_1, j_2, u_1
  33. Polygon(R, U, F, T) → q4, r_5, u_2, f_2, t_2
  34. Translate(q1, Vector[R, If[2 < u ≤ 3, Dilate[R, 3 - u, A], A]]) → q1'
  35. Translate(q2, Vector[R, If[4 < u ≤ 5, Dilate[R, 5 - u, O], O]]) → q2'
  36. Translate(q3, Vector[R, If[6 < u ≤ 7, Dilate[R, 7 - u, P], P]]) → q3'
  37. Translate(q4, Vector[R, If[8 < u ≤ 9, Dilate[R, 9 - u, C], C]]) → q4'
  38. Midpoint(B, L) → Z
  39. Midpoint(M, B) → A_1
  40. Translate(poly3, Vector[Z, If[10 < u ≤ 11, Dilate[Z, 11 - u, A_1], A_1]]) → poly3'
  41. Circle(f_1, B) → d_1
  42. Intersect(xOyPlane, d_1) → B_1, C_1
  43. Segment(C, C_1) → b_1
  44. OrthogonalPlane(S, b_1) → p_2
  45. OrthogonalLine(S, p_2) → e_1
  46. OrthogonalLine(S, xOyPlane) → g_2
  47. Plane(g_2, e_1) → q_2
  48. OrthogonalLine(S, q_2) → h_2
  49. Vector(R, O) → v_2
  50. Vector(R, P) → w_2
  51. Vector(R, C) → a_2
  52. Vector(R, A) → b_2
  53. Intersect(h_2, poly5) → D_1, E_1
  54. Intersect(e_1, poly5) → F_1, G_1
  55. Polygon(S, G_1, A, D_1) → q5, s_2, k_2, a_3, l_2
  56. Polygon(S, D_1, C, F_1) → q6, s_3, m_2, c_3, n_2
  57. Polygon(S, F_1, P, E_1) → q7, s_4, d_2, p_3, e_2
  58. Polygon(S, E_1, O, G_1) → q8, s_5, f_3, o, g_3
  59. Rotate(S, ((-π)) / 2, yAxis) → S'
  60. Translate(F, a_2) → F'
  61. Translate(C, b_2) → C'
  62. Translate(I, v_2) → I'
  63. Translate(J, w_2) → J'
  64. Midpoint(O, A) → H_1
  65. Midpoint(A, C) → I_1
  66. Midpoint(C, P) → J_1
  67. Midpoint(P, O) → K_1
  68. Segment(C', F') → h_3
  69. Segment(F', J') → i_3
  70. Segment(J', I') → j_3
  71. Segment(I', C') → k_3
  72. Intersect(h_3, g_3) → L_1
  73. Intersect(s_3, i_3) → M_1
  74. Intersect(j_3, s_4) → N_1
  75. Intersect(k_3, s_5) → O_1
  76. Polygon(S, L_1, F', M_1) → q9, s_6, l_3, f', m_3
  77. Polygon(L_1, G_1, A, I_1) → q10, n_3, q_3, a_4, t_3
  78. Polygon(M_1, F', I_1, D_1) → q11, b_3, d_3, e_3, f_4
  79. Translate(If[12 < u ≤ 13, Rotate[q9, ((-(u - 12)) * π / 2), yAxis], Rotate[q9, ((-π)) / 2, yAxis]], Vector[S', If[12 < u ≤ 13, Dilate[S', 13 - u, C], C]]) → q13
  80. Translate(If[12 < u ≤ 13, Rotate[q10, ((-(u - 12)) * π / 2), yAxis], Rotate[q10, ((-π)) / 2, yAxis]], Vector[S', If[12 < u ≤ 13, Dilate[S', 13 - u, C], C]]) → q14
  81. Translate(If[12 < u ≤ 13, Rotate[q11, ((-(u - 12)) * π / 2), yAxis], Rotate[q11, ((-π)) / 2, yAxis]], Vector[S', If[12 < u ≤ 13, Dilate[S', 13 - u, C], C]]) → q15
  82. Polygon(N_1, S, M_1, J') → q12, g_4, s_7, h_4, j'
  83. Polygon(M_1, D_1, C, J_1) → q16, i_4, j_4, c_4, k_4
  84. Polygon(F_1, N_1, J', J_1) → q17, l_4, m_4, n_4, p_4
  85. Translate(If[14 < u ≤ 15, Rotate[q12, ((-(u - 14)) * π / 2), yAxis], Rotate[q12, ((-π)) / 2, yAxis]], Vector[S', If[14 < u ≤ 15, Dilate[S', 15 - u, B], B]]) → q18
  86. Translate(If[14 < u ≤ 15, Rotate[q16, ((-(u - 14)) * π / 2), yAxis], Rotate[q16, ((-π)) / 2, yAxis]], Vector[S', If[14 < u ≤ 15, Dilate[S', 15 - u, B], B]]) → q19
  87. Translate(If[14 < u ≤ 15, Rotate[q17, ((-(u - 14)) * π / 2), yAxis], Rotate[q17, ((-π)) / 2, yAxis]], Vector[S', If[14 < u ≤ 15, Dilate[S', 15 - u, B], B]]) → q20
  88. Polygon(I', O_1, S, N_1) → q21, i', o_1, s_8, q_4
  89. Polygon(K_1, N_1, F_1, P) → q22, t_4, b_4, d_4, p_5
  90. Polygon(I', K_1, E_1, O_1) → q23, e_4, f_5, g_5, h_5
  91. Translate(If[16 < u ≤ 17, Rotate[q21, ((-(u - 16)) * π / 2), yAxis], Rotate[q21, (-π) / 2, yAxis]], Vector[S', If[16 < u ≤ 17, Dilate[S', 17 - u, N], N]]) → q24
  92. Translate(If[16 < u ≤ 17, Rotate[q22, ((-(u - 16)) * π / 2), yAxis], Rotate[q22, (-π) / 2, yAxis]], Vector[S', If[16 < u ≤ 17, Dilate[S', 17 - u, N], N]]) → q25
  93. Translate(If[16 < u ≤ 17, Rotate[q23, ((-(u - 16)) * π / 2), yAxis], Rotate[q23, (-π) / 2, yAxis]], Vector[S', If[16 < u ≤ 17, Dilate[S', 17 - u, N], N]]) → q26
  94. Polygon(O_1, C', L_1, S) → q27, i_5, c', j_5, s_9
  95. Polygon(O_1, E_1, O, H_1) → q28, k_5, l_5, o_2, m_5
  96. Polygon(C', H_1, G_1, L_1) → q29, n_5, q_5, t_5, a_5
  97. Translate(If[18 < u ≤ 19, Rotate[q27, ((-(u - 18)) * π / 2), yAxis], Rotate[q27, (-π) / 2, yAxis]], Vector[S', If[18 < u ≤ 19, Dilate[S', 19 - u, M], M]]) → q30
  98. Translate(If[18 < u ≤ 19, Rotate[q28, ((-(u - 18)) * π / 2), yAxis], Rotate[q28, (-π) / 2, yAxis]], Vector[S', If[18 < u ≤ 19, Dilate[S', 19 - u, M], M]]) → q31
  99. Translate(If[18 < u ≤ 19, Rotate[q29, ((-(u - 18)) * π / 2), yAxis], Rotate[q29, (-π) / 2, yAxis]], Vector[S', If[18 < u ≤ 19, Dilate[S', 19 - u, M], M]]) → q32
  100. Angle(B, A, C) → α
  101. Angle(A, F, C) → β