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通过面积发现勾股定理

通过面积发现勾股定理
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2015-08-09 02:28
本范例通过在网格上寻找斜边平方的面积来演示勾股定理。涉及的核心数学概念是勾股定理、面积计算、几何变换和比例关系。作图要点包括使用Sequence、Union、Point、If、Translate、OrthogonalLine、Segment、Midpoint、Intersect、Ray、Circle、PointIn、Polygon、Defined、Mirror、Textfield、Vector、Angle等GeoGebra命令和对象。适合初中及以上学生用于理解勾股定理的直观证明和几何应用。

GeoGebra 命令及参数

  1. Sequence((i, 0), i, 0, 20) → L0
  2. Sequence((i, 1), i, 0, 20) → L1
  3. Sequence((i, 2), i, 0, 20) → L2
  4. Sequence((i, 3), i, 0, 20) → L3
  5. Sequence((i, 4), i, 0, 20) → L4
  6. Sequence((i, 5), i, 0, 20) → L5
  7. Sequence((i, 6), i, 0, 20) → L6
  8. Sequence((i, 7), i, 0, 20) → L7
  9. Sequence((i, 8), i, 0, 20) → L8
  10. Sequence((i, 9), i, 0, 20) → L9
  11. Sequence((i, 10), i, 0, 20) → L10
  12. Sequence((i, 11), i, 0, 20) → L11
  13. Sequence((i, 12), i, 0, 20) → L12
  14. Sequence((i, 13), i, 0, 20) → L13
  15. Sequence((i, 14), i, 0, 20) → L14
  16. Sequence((i, 15), i, 0, 20) → L15
  17. Sequence((i, 16), i, 0, 20) → L16
  18. Sequence((i, 17), i, 0, 20) → L17
  19. Sequence((i, 18), i, 0, 20) → L18
  20. Sequence((i, 19), i, 0, 20) → L19
  21. Sequence((i, 20), i, 0, 20) → L20
  22. Union(L1, Union[L2, Union[L3, Union[L4, Union[L5, Union[L6, Union[L7, Union[L8, Union[L9, Union[L10, Union[L11, Union[L12, Union[L13, Union[L14, Union[L15, Union[L16, Union[L17, Union[L18, Union[L19, Union[L20, L0]]]]]]]]]]]]]]]]]]]) → L'
  23. Point(L') → B'
  24. Point(L') → A'
  25. If(a' * b' ≥ 0, a, b) → x1
  26. If(a' * b' ≥ 0, b, a) → y1
  27. Translate(Midpoint[A, B], Vector[(0, 0), (0, 1)]) → J
  28. OrthogonalLine(J, yAxis) → c
  29. Segment(B, A) → d
  30. Midpoint(A, B) → I
  31. OrthogonalLine(I, d) → e
  32. Intersect(c, e) → K
  33. Ray(I, K) → f
  34. Circle(I, A) → g
  35. Intersect(g, f, 1) → O
  36. PointIn(Polygon[E, (x(E), y(E) + x1), (x(E) - y1, y(E) + x1), (x(E) - y1, y(E))]) → P
  37. Polygon(P, (x(P), y(P) + y1), (x(P) - x1, y(P))) → poly2, j, p, k
  38. Translate(poly2, z') → poly2_1
  39. PointIn(Polygon[F, (x(F) - x1, y(F)), (x(F) - x1, y(F) - y1), (x(F), y(F) - y1)]) → Q
  40. Polygon(Q, (x(Q) - y1, y(Q)), (x(Q), y(Q) - x1)) → poly1, h, q, i
  41. Translate(poly1, z') → poly1_1
  42. Defined(IntersectRegion[poly1_1, poly2_1]) → overlapPQ
  43. PointIn(Polygon[G, (x(G), y(G) - x1), (x(G) + y1, y(G) - x1), (x(G) + y1, y(G))]) → R
  44. Polygon(R, (x(R), y(R) - y1), (x(R) + x1, y(R))) → poly3, l, r_1, m
  45. Translate(poly3, z') → poly3_1
  46. Defined(IntersectRegion[poly3_1, poly1_1]) → overlapQR
  47. Defined(IntersectRegion[poly3_1, poly2_1]) → overlapPR
  48. PointIn(Polygon[H, (x(H) + x1, y(H)), (x(H) + x1, y(H) + y1), (x(H), y(H) + y1)]) → S
  49. Polygon(S, (x(S) + y1, y(S)), (x(S), y(S) + x1)) → poly4, n, s, r
  50. Translate(poly4, z') → poly4_1
  51. Defined(IntersectRegion[poly4_1, poly3_1]) → overlapRS
  52. Defined(IntersectRegion[poly4_1, poly1_1]) → overlapQS
  53. Defined(IntersectRegion[poly4_1, poly2_1]) → overlapPS
  54. Mirror((x(A), y(A)), O) → C
  55. Mirror((x(B), y(B)), O) → D
  56. Polygon(E, F, G, H) → poly5, e_1, f_1, g_1, h_1
  57. Polygon(A, B, C, D) → poly6, t, a_1, b_1, c_1
  58. Segment((x(R), y(R)), (x(R), y(R) + x1)) → d1
  59. Segment((x(Q), y(Q)), (x(Q) + x1, y(Q))) → d2
  60. Segment((x(P), y(P)), (x(P), y(P) - x1)) → d3
  61. Segment((x(S), y(S)), (x(S) - x1, y(S))) → d4
  62. Segment((x(Q), y(Q)), (x(Q) - a - b, y(Q))) → d5
  63. Segment((x(S), y(S)), (x(S) + a + b, y(S))) → d6
  64. Segment((x(R), y(R)), (x(R), y(R) - a - b)) → d7
  65. Segment((x(P), y(P)), (x(P), y(P) + a + b)) → d8
  66. Sequence(Segment[(i, y(E)), (i, y(F))], i, x(H), x(E)) → list1
  67. Sequence(Segment[(x(H), j), (x(E), j)], j, y(E), y(F)) → list2
  68. Textfield(Ar) → inputBox1
  69. Midpoint(A, E) → L
  70. Midpoint(B, E) → M
  71. Midpoint(A, H) → N
  72. Midpoint(B, H) → T
  73. Midpoint(A', B') → I'
  74. Circle(I', A') → g'
  75. Segment(B', A') → d'
  76. OrthogonalLine(I', d') → e'
  77. Translate(Midpoint[A', B'], Vector[(0, 0), (0, 1)]) → J'
  78. OrthogonalLine(J', yAxis) → c'
  79. Intersect(c', e') → K'
  80. Ray(I', K') → f'
  81. Intersect(g', f', 1) → O'
  82. Sequence(Segment[(i, y(E')), (i, y(F'))], i, x(H'), x(E')) → list3
  83. Sequence(Segment[(x(H'), j), (x(E'), j)], j, y(E'), y(F')) → list4
  84. Mirror((x(A'), y(A')), O') → C'
  85. Mirror((x(B'), y(B')), O') → D'
  86. Polygon(E', (x(E'), y(E') + y1), (x(E') - x1, y(E'))) → poly1', d_1, e'_1, i_1
  87. Polygon(F', (x(F') - y1, y(F')), (x(F'), y(F') - x1)) → poly2', j_1, f'_1, k_1
  88. Polygon(G', (x(G'), y(G') - y1), (x(G') + x1, y(G'))) → poly3', l_1, g'_1, m_1
  89. Polygon(H', (x(H') + y1, y(H')), (x(H'), y(H') + x1)) → poly4', n_1, h', p_1
  90. Polygon(A', B', C', D') → poly6', a'_1, b'_1, c'_1, d'_1
  91. Vector(A', E') → u
  92. Vector(E', A') → v
  93. Vector(B', E') → w
  94. Vector(E', B') → z
  95. Midpoint(e'_1) → C_1
  96. Midpoint(f'_1) → D_1
  97. Midpoint(g'_1) → E_1
  98. Midpoint(h') → F_1
  99. Midpoint(p) → G_1
  100. Midpoint(q) → H_1
  101. Midpoint(r_1) → I_1
  102. Midpoint(s) → J_1
  103. Segment(E, F) → q_1
  104. Segment(F, G) → s_1
  105. Segment(G, H) → t_1
  106. Segment(H, E) → a_2
  107. Angle((x(E') - x1, y(E')), E', (x(E'), y(E') + y1)) → α
  108. Angle((x(P) - x1, y(P)), P, (x(P), y(P) + y1)) → α'
  109. Angle((x(F') - y1, y(F')), F', (x(F'), y(F') - x1)) → β
  110. Angle((x(Q) - y1, y(Q)), Q, (x(Q), y(Q) - x1)) → β'
  111. Angle((x(G'), y(G') - y1), G', (x(G') + x1, y(G'))) → γ
  112. Angle((x(R), y(R) - y1), R, (x(R) + x1, y(R))) → γ'
  113. Angle((x(S), y(S) + x1), S, (x(S) + y1, y(S))) → δ'
  114. Angle((x(H'), y(H') + x1), H', (x(H') + y1, y(H'))) → δ
  115. If(show1 ∨ show9 ∨ show6, Midpoint[F, (x(F) - a, y(F) - a)], If[show10 ∨ show4 ∨ show8, Midpoint[G, (x(G) + a, y(G) - a)], If[show3 ∨ show5 ∨ show11, Midpoint[H, (x(H) + a, y(H) + a)], If[show2 ∨ show12 ∨ show7, Midpoint[E, (x(E) - a, y(E) + a)]]]]) → K_1
  116. If(show5 ∨ show12 ∨ show1, Midpoint[G, (x(G) + b, y(G) - b)], If[show2 ∨ show11 ∨ show8, Midpoint[F, (x(F) - b, y(F) - b)], If[show3 ∨ show10 ∨ show6, Midpoint[E, (x(E) - b, y(E) + b)], If[show9 ∨ show4 ∨ show7, Midpoint[H, (x(H) + b, y(H) + b)]]]]) → K_2
  117. Midpoint(E', G') → L_1
  118. Midpoint(E, G) → M_1

AI 绘图提示语参考

在一个网格上，画出20条水平线和20条竖直线，它们的交点构成一个20x20的正方形。接着，在这个正方形内部，画出四个斜放的正方形，分别位于正方形的四个角落，并计算它们的面积。通过几何变换和辅助线，展示这些斜放正方形的面积之和等于正方形内部两个大正方形的面积之和，从而证明勾股定理。
以上提示语为根据原始绘图命令整理而来，仅供参考。