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GeoGebra几何探究：半分

GeoGebra几何探究：半分
几何画板及绘图内容加载中，请稍等……
2023-03-22 03:12
测试改名0圆多边形几何作图
所属专题 Realidades de GeoGebra
本范例演示了在GeoGebra中通过几何作图探究图形的半分问题。涉及的核心数学概念包括面积、构造、分部、正方形、微分计算、三角形、复数等。作图要点使用了Intersect（相交）、Point（点）、Line（线）、OrthogonalLine（垂线）、Circle（圆）、Polygon（多边形）、Vector（向量）、Translate（平移）、Segment（线段）、Midpoint（中点）、Ray（射线）等命令和对象。适合高中数学教学或几何作图学习。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Line(A, B) → a
  4. OrthogonalLine(A, a) → b
  5. Circle(A, B) → c
  6. Intersect(c, b, 2) → C
  7. OrthogonalLine(C, b) → d
  8. OrthogonalLine(B, a) → e
  9. Intersect(d, e) → D
  10. Polygon(A, B, D, C) → poly1, a_1, b_1, d_1, c_1
  11. Line(E, a_1) → f
  12. Point(f) → F
  13. Vector(E, F) → u
  14. OrthogonalLine(E, f) → g
  15. Circle(E, F) → h
  16. Intersect(h, g, 2) → G
  17. Vector(E, G) → v
  18. Translate(A, u) → A'
  19. Translate(B, u) → B'
  20. Translate(D, u) → D'
  21. Translate(C, u) → C'
  22. Polygon(A', B', D', C') → poly1', a', b', d', c'
  23. Translate(A', u) → A''
  24. Translate(B', u) → B''
  25. Translate(D', u) → D''
  26. Translate(C', u) → C''
  27. Polygon(A'', B'', D'', C'') → poly1'', a'', b'', d'', c''
  28. Translate(A, v) → A'_1
  29. Translate(B, v) → B'_1
  30. Translate(D, v) → D'_1
  31. Translate(C, v) → C'_1
  32. Polygon(A'_1, B'_1, D'_1, C'_1) → poly1'_1, a'_1, b'_1, d'_1, c'_1
  33. Translate(A', v) → A''_1
  34. Translate(B', v) → B''_1
  35. Translate(D', v) → D''_1
  36. Translate(C', v) → C''_1
  37. Polygon(A''_1, B''_1, D''_1, C''_1) → poly1''_1, a''_1, b''_1, d''_1, c''_1
  38. Translate(A'', v) → A'''
  39. Translate(B'', v) → B'''
  40. Translate(D'', v) → D'''
  41. Translate(C'', v) → C'''
  42. Polygon(A''', B''', D''', C''') → poly1''', a''', b''', d''', c'''
  43. Translate(D'_1, v) → D'_1'
  44. Translate(C'_1, v) → C'_1'
  45. Segment(D'_1', C'_1') → i
  46. Translate(D'_1, v) → D'_2
  47. Translate(C'_1, v) → C'_2
  48. Segment(D'_2, C'_2) → j
  49. Translate(A'_1, v) → A'_1'
  50. Translate(B'_1, v) → B'_1'
  51. Translate(D'_1, v) → D'_3
  52. Translate(C'_1, v) → C'_3
  53. Polygon(A'_1', B'_1', D'_3, C'_3) → poly1'_1', a'_1', b'_1', d'_3, c'_3
  54. Translate(A''_1, v) → A''_1'
  55. Translate(B''_1, v) → B''_1'
  56. Translate(D''_1, v) → D''_1'
  57. Translate(C''_1, v) → C''_1'
  58. Polygon(A''_1', B''_1', D''_1', C''_1') → poly1''_1', a''_1', b''_1', d''_1', c''_1'
  59. Translate(A''', v) → A''''
  60. Translate(B''', v) → B''''
  61. Translate(D''', v) → D''''
  62. Translate(C''', v) → C''''
  63. Polygon(A'''', B'''', D'''', C'''') → poly1'''', a'''', b'''', d'''', c''''
  64. Translate(A'', u) → A'''_1
  65. Translate(B'', u) → B'''_1
  66. Translate(D'', u) → D'''_1
  67. Translate(C'', u) → C'''_1
  68. Polygon(A'''_1, B'''_1, D'''_1, C'''_1) → poly1'''_1, a'''_1, b'''_1, d'''_1, c'''_1
  69. Translate(A''', u) → A''''_1
  70. Translate(B''', u) → B''''_1
  71. Translate(D''', u) → D''''_1
  72. Translate(C''', u) → C''''_1
  73. Polygon(A''''_1, B''''_1, D''''_1, C''''_1) → poly1''''_1, a''''_1, b''''_1, d''''_1, c''''_1
  74. Translate(A'''', u) → A'''''
  75. Translate(B'''', u) → B'''''
  76. Translate(D'''', u) → D'''''
  77. Translate(C'''', u) → C'''''
  78. Polygon(A''''', B''''', D''''', C''''') → poly1''''', a''''', b''''', d''''', c'''''
  79. Midpoint(i) → H
  80. Midpoint(a'_1') → I
  81. Polygon(H, D'_1', B'_1', I) → poly2, h_1, d'_1', b'_2, i_1
  82. Midpoint(d''_1') → J
  83. Segment(C''_1', J) → k
  84. Point(k) → K
  85. Circle(K, m) → p
  86. Intersect(p, d''_1', 1) → L
  87. OrthogonalLine(K, k) → l
  88. OrthogonalLine(L, d''_1') → n
  89. Intersect(a''_1', l) → M
  90. Intersect(a''_1', n) → N
  91. Polygon(M, N, L, K) → poly3, m_1, n_1, l_1, k_1
  92. Midpoint(d'_1) → O
  93. Polygon(A'_1, O, B'_1) → poly4, b'_3, a'_2, o
  94. Point(d''_1) → P
  95. Polygon(A''_1, P, B''_1) → poly5, b''_2, a''_2, p_1
  96. Midpoint(d_1) → Q
  97. Midpoint(a_1) → R
  98. Polygon(A, Q, D, R) → poly6, a_2, q, d_2, r
  99. Midpoint(d') → S
  100. Midpoint(a') → T
  101. Midpoint(a_1) → U
  102. Segment(C', S) → s
  103. Segment(T, A') → t
  104. Point(s) → V
  105. Point(t) → W
  106. Circle(V, m) → e_1
  107. Circle(W, m) → f_1
  108. Polygon(W, V, Z, A_1) → poly7, w, v_1, z, a_3
  109. Midpoint(D'', A'') → E_1
  110. Midpoint(A'', E_1) → F_1
  111. Midpoint(E_1, D'') → G_1
  112. Segment(C'', B'') → j_1
  113. Polygon(C'', F_1, B'', G_1) → poly8, c''_2, f_2, b''_3, g_2
  114. Midpoint(d'''_1) → H_1
  115. Midpoint(b'''_1) → I_1
  116. Midpoint(a'''_1) → J_1
  117. Midpoint(c'''_1) → K_1
  118. Segment(K_1, J_1) → q_1
  119. Segment(I_1, H_1) → r_1
  120. Point(q_1) → L_1
  121. Point(r_1) → M_1
  122. Polygon(C'''_1, L_1, B'''_1, M_1) → poly9, c'''_2, l_2, b'''_2, m_2
  123. Midpoint(d'''') → B_1
  124. Midpoint(c'''') → C_1
  125. Midpoint(a'''') → D_1
  126. Midpoint(b'''') → N_1
  127. Polygon(B_1, N_1, D_1, C_1) → poly10, b_2, n_2, d_3, c_2
  128. Point(d''''') → O_1
  129. Point(c''''') → P_1
  130. OrthogonalLine(P_1, c''''') → g_1
  131. OrthogonalLine(O_1, d''''') → s_1
  132. Intersect(a''''', s_1) → Q_1
  133. Intersect(b''''', g_1) → R_1
  134. Polygon(P_1, Q_1, R_1, O_1) → poly11, p_2, q_2, r_2, o_1
  135. Midpoint(d''') → S_1
  136. Midpoint(a''') → T_1
  137. Midpoint(S_1, D''') → U_1
  138. Midpoint(T_1, A''') → V_1
  139. Polygon(U_1, V_1, B''', D''') → poly12, u_1, v_2, b'''_3, d'''_2
  140. Point(d''''_1) → W_1
  141. Midpoint(A''''_1, D''''_1) → Z_1
  142. Ray(W_1, Z_1) → t_1
  143. Intersect(a''''_1, t_1) → A_2
  144. Polygon(A_2, B''''_1, D''''_1, W_1) → poly13, a_4, b''''_2, d''''_2, w_1

AI 绘图提示语参考

在坐标平面上，以原点为一个顶点，在横轴上取一点作为另一顶点，构造一个直角三角形。以原点为圆心、到该点的距离为半径画圆，与过原点且垂直于斜边的直线相交，得到第三个顶点，再补全为正方形。随后，在正方形外构造一条射线，并在其上取一点，以该点为方向向量，将整个正方形沿此方向平移两次，形成一排三个正方形。再以垂直方向为另一平移向量，将这一排正方形整体平移两次，形成三排共九个正方形的网格。最后，在网格边缘连接两个平移后的顶点，画出两条平行线段，展示分割与对半的几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。