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正方形内四分之一圆与动态几何关系（GoGeometry 第81题）

正方形内四分之一圆与动态几何关系（GoGeometry 第81题）
几何画板及绘图内容加载中，请稍等……
2017-07-28 07:49
圆多边形变换几何作图滑块动画
该范例演示了在正方形内部绘制一个四分之一圆（扇形）时，所形成的一系列动态且可修改的几何关系。核心数学概念包括圆与扇形的性质、正方形与多边形的边长关系、旋转与缩放变换、以及中点、垂线、角平分等基本几何元素。作图要点：使用GeoGebra的Ray、Segment、Circle、Intersect、Point、OrthogonalLine、Line、Polygon、Rotate、Dilate、Midpoint、CircleArc、CircleSector、Vector、Translate、Angle、Mirror等命令，构建了包含点A、B、C、D、多边形poly1、圆弧与扇形、以及多条辅助线和变换后对象（如A'、A''等）的复杂构造。该范例适合中学几何教学或数学爱好者探索动态几何关系，可用于理解圆内接图形、对称变换和比例关系。

GeoGebra 命令及参数

  1. Ray(A, B_2) → m_2
  2. Segment(A, B_2) → g
  3. Circle(B_2, g) → q_2
  4. Intersect(q_2, m_2, 2) → J_1
  5. Segment(B_2, J_1) → n_2
  6. Point(n_2) → K_1
  7. Segment(B_2, K_1) → r_2
  8. OrthogonalLine(K_1, r_2) → s_2
  9. Circle(K_1, r_2) → t_2
  10. Intersect(t_2, s_2, 2) → L_1
  11. Line(K_1, L_1) → h_3
  12. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  13. Segment(B_2, C) → h
  14. OrthogonalLine(L_1, h) → f_3
  15. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  16. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  17. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  18. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  19. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  20. Segment(R, A''') → f''
  21. Midpoint(A, C) → V
  22. CircleArc(V, C, D) → g_2
  23. Point(g_2) → B
  24. Segment(C, B) → j_1
  25. Segment(A, B) → n
  26. Intersect(n, f'') → Z
  27. Segment(B, Z) → l_1
  28. Segment(A, A') → f
  29. Segment(A'_1, A'') → f'
  30. Segment(C, D) → i
  31. Circle(B_2, (0.2 * r_2)) → p
  32. Segment(D, A) → j
  33. Circle(C, Radius[p]) → k
  34. Circle(A, (0.1 * j)) → q
  35. Circle(D, (0.1 * j)) → r
  36. Segment(A, C) → l
  37. Segment(D, B_2) → m
  38. Intersect(r, j, 1) → F
  39. Intersect(r, m, 1) → G
  40. Intersect(r, i, 1) → H
  41. Intersect(k, i, 1) → I
  42. Intersect(k, l, 1) → J
  43. Intersect(k, h, 1) → K
  44. Intersect(p, h, 1) → L
  45. Intersect(p, m, 1) → M
  46. Intersect(p, g, 1) → N
  47. Intersect(q, g, 1) → O
  48. Intersect(q, l, 1) → P
  49. Intersect(q, j, 1) → Q
  50. CircleSector(B_2, L, N) → s
  51. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  52. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  53. Segment(R', S) → f'''
  54. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  55. Vector(B_2, B') → u
  56. Translate(s, u) → s'
  57. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  58. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  59. Vector(C, C') → v
  60. Translate(s'', v) → s'''
  61. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  62. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  63. Vector(D, D') → w
  64. Translate(t, w) → t'
  65. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  66. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  67. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  68. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  69. Vector(B_2, B'_1) → v_1
  70. Vector(B_2, B'_3) → w_1
  71. Vector(B_2, B'_2) → i_1
  72. Circle(A, n - j_1) → h_2
  73. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  74. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  75. Vector(B_2, B'_4) → u_1
  76. Translate(s, u_1) → s'_1
  77. Line(B, f') → a_1
  78. Point(a_1) → U
  79. Angle(U, B, C) → α
  80. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  81. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  82. Line(A, B_2) → h_1
  83. Angle(B, A, D) → β
  84. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  85. Mirror(C, D) → C'_1
  86. Rotate(C, ((-π)) / 3, B_2) → C'_2
  87. Segment(B_2, C'_2) → f_1
  88. Segment(C, C'_2) → g_1
  89. Intersect(k, g_1, 1) → E
  90. Intersect(k, h) → T,
  91. CircleSector(C, T, E) → d_1
  92. Rotate(D, 15°, C) → D'_2
  93. Angle(D, C, D'_2) → γ
  94. Ray(C, D'_2) → k_1
  95. Rotate(C, (-(15°)), D) → C'_3
  96. Angle(C'_3, D, C) → δ
  97. Ray(D, C'_3) → m_1
  98. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  99. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  100. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  101. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  102. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  103. Rotate(T, 15°, C) → T'
  104. Angle(T, C, T') → ε
  105. Ray(C, T') → q_1
  106. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  107. Vector(C, C'_8) → u_2
  108. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  109. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  110. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  111. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  112. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  113. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  114. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  115. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  116. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  117. Vector(B_2, Q_1) → c_1
  118. Segment(C, K_1) → e_1
  119. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  120. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  121. Line(A, K_1) → k_3
  122. Intersect(t, f'', 1) → G_2
  123. Rotate(B_2, If[10 < e ≤ 11, ((e - 10) * π / 2), π / 2], A) → W
  124. CircleArc(A, B_2, W) → p_1
  125. Rotate(A, If[10 < e ≤ 11, ((e - 10) * π / 2), π / 2], A) → A'_2
  126. Rotate(A', If[10 < e ≤ 11, ((e - 10) * π / 2), π / 2], A) → A''_1
  127. Segment(A'_2, A''_1) → f'_1
  128. Segment(A, C) → n_1
  129. Segment(B_2, D) → r_1
  130. Intersect(r_1, n_1) → G_1
  131. Intersect(p_1, n_1) → M_1,
  132. Line(B_2, M_1) → s_1
  133. Intersect(s_1, f'') → O_1
  134. Dilate(D, If[11 < e ≤ 12, 12 - e, 0], B_2) → D'_3
  135. Segment(D, D'_3) → t_1
  136. Dilate(A, If[12 < e ≤ 13, 13 - e, 0], G_1) → A'_3
  137. Segment(A, A'_3) → b_1
  138. Dilate(G_1, If[13 < e ≤ 14, 14 - e, 0], M_1) → P_1
  139. Segment(G_1, P_1) → f_2
  140. Dilate(M_1, If[14 < e ≤ 15, 15 - e, 0], C) → R_1
  141. Segment(M_1, R_1) → i_2
  142. Dilate(B_2, If[15 < e ≤ 16, 16 - e, 0], M_1) → S_1
  143. Segment(B_2, S_1) → j_2
  144. Dilate(M_1, If[16 < e ≤ 17, 17 - e, 0], O_1) → T_1
  145. Segment(M_1, T_1) → k_2
  146. Dilate(C, If[16 < e ≤ 17, 17 - e, 0], O_1) → C'_7
  147. Segment(C, C'_7) → l_2
  148. Midpoint(D, O_1) → U_1
  149. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], O_1) → V_1
  150. Vector(M_1, V_1) → v_2
  151. Translate(G_1, v_2) → W_1
  152. Translate(P_1, v_2) → Z_1
  153. Segment(W_1, Z_1) → p_2
  154. Rotate(W_1, If[18 < e ≤ 19, ((-(e - 18)) * π / 4), ((-π)) / 4], O_1) → A_2
  155. Rotate(Z_1, If[18 < e ≤ 19, ((-(e - 18)) * π / 4), ((-π)) / 4], O_1) → C_2
  156. Segment(A_2, C_2) → a_2
  157. Rotate(A_2, If[19 < e ≤ 20, ((-(e - 19)) * π), (-π)], U_1) → D_2
  158. Rotate(C_2, If[19 < e ≤ 20, ((-(e - 19)) * π), (-π)], U_1) → E_2
  159. Segment(D_2, E_2) → b_2
  160. CircleSector(A, B_2, W) → c_2