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菱形与内切圆的动态演示

菱形与内切圆的动态演示
几何画板及绘图内容加载中，请稍等……
2017-03-10 11:36
圆多边形三角函数几何作图
所属专题 Polygons & Angles Polygons & Angles ( s n attar) Polygons & Angles
①本范例动态演示了菱形及其内切圆的几何关系；②涉及核心数学概念包括菱形的对称性、内切圆性质、角度计算、几何变换等；③构造过程中运用了Line（线段）、Circle（圆）、Intersect（交点）、Rotate（旋转）、Dilate（缩放）等GeoGebra命令与对象；④适合高中数学教学，用于理解几何变换与图形性质，提升空间想象能力。

GeoGebra 命令及参数

  1. Line(A, B) → b
  2. Circle(A, B) → c
  3. Intersect(c, b, 2) → C
  4. CircleArc(A, B, C) → d
  5. Point(d) → D
  6. Circle(D, A) → e
  7. Segment(D, B) → s
  8. Circle(B, A) → f
  9. Intersect(e, f, 2) → E
  10. Angle(D, E, B) → β
  11. Angle(E, B, A) → α
  12. Segment(A, E) → n
  13. Intersect(n, s) → L
  14. Rotate(L, If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L'_2
  15. Dilate(L, If[12 < l ≤ 13, 13 - l, 0], E) → L'_1
  16. Segment(L, L'_1) → j_1
  17. Dilate(L, If[10 < l ≤ 11, 11 - l, 0], B) → L'
  18. Rotate(L', If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L''
  19. Segment(L'_2, L'') → l_1
  20. Rotate(B, If[1 < l ≤ 2, ((-(l - 1)) * α), α], B) → B'_1
  21. Rotate(B'_1, If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → J
  22. Rotate(J, If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → J'
  23. Dilate(B, If[0 < l ≤ 1, 1 - l, 0], A) → B'
  24. Rotate(B', If[1 < l ≤ 2, ((-(l - 1)) * α), (-α)], B) → B''
  25. Rotate(B'', If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → B'''
  26. Rotate(B''', If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → K
  27. Segment(J', K) → m'''
  28. Segment(J, B''') → m''
  29. OrthogonalLine(L, m'') → n_1
  30. Intersect(n_1, m'') → B_1
  31. Circle(L, B_1) → g_2
  32. Intersect(g_2, j_1) → I_1,
  33. Intersect(g_2, l_1) → H_1,
  34. Intersect(g_2, m''', 1) → C_1
  35. Segment(C_1, I_1) → a_2
  36. Segment(B'_1, B'') → m'
  37. Intersect(g_2, m', 1) → E_1
  38. Segment(H_1, E_1) → t_2
  39. Intersect(a_2, t_2) → Q_1
  40. Segment(A, B) → a
  41. Segment(A, D) → g
  42. Segment(D, E) → h
  43. Segment(E, B) → i
  44. Min(g, h) → j
  45. Segment(B, B') → m
  46. Circle(B, (0.2 * j)) → k
  47. Circle(E, (0.2 * j)) → p
  48. Intersect(p, m'', 1) → F
  49. Intersect(p, m', 1) → G
  50. Intersect(k, m', 1) → H
  51. Intersect(k, m, 1) → I
  52. CircleSector(B, H, I) → q
  53. CircleSector(E, F, G) → r
  54. Polygon(A, B, E, D) → poly1, a_1, b_1, e_1, d_1
  55. Dilate(B, If[5 < l ≤ 6, 6 - l, 0], D) → B'_2
  56. Vector(B, B'_2) → u
  57. Translate(q, u) → q'
  58. Rotate(q', If[6 < l ≤ 7, ((l - 6) * π), 3.141592653589793], D) → q''
  59. Dilate(E, If[7 < l ≤ 8, 8 - l, 0], A) → E'
  60. Vector(E, E') → v
  61. Translate(r, v) → r'
  62. Rotate(r', If[8 < l ≤ 9, ((-(l - 8)) * π), -3.141592653589793], A) → r''
  63. Dilate(D, If[9 < l ≤ 10, 10 - l, 0], L) → D'
  64. Vector(D, D') → w
  65. Min(n / 2, s / 2) → o
  66. Circle(L, (0.2 * o)) → t
  67. Intersect(t, s, 2) → M
  68. Intersect(t, n, 2) → N
  69. CircleSector(L, M, N) → c_1
  70. Segment(D, D') → f_1
  71. Circle(D, (0.2 * j)) → h_1
  72. Intersect(k, s) → P,
  73. Intersect(s, f_1) → Q
  74. Intersect(h_1, s, 1) → O
  75. Circle(A, (0.2 * j)) → k_1
  76. Intersect(k_1, m, 1) → R
  77. Intersect(k_1, n, 1) → S
  78. Intersect(k_1, m''', 1) → T
  79. Intersect(h_1, m'', 1) → U
  80. CircleSector(D, O, U) → q_1
  81. Dilate(A, If[11 < l ≤ 12, 12 - l, 0], L) → A'
  82. Segment(A, A') → i_1
  83. Intersect(p, m'', 1) → V
  84. Intersect(p, n, 1) → W
  85. CircleSector(A, S, T) → s_1
  86. CircleSector(B, P, I) → p_1
  87. Intersect(p, m', 1) → Z
  88. CircleSector(E, W, Z) → r_1
  89. Segment(L, L') → g_1
  90. Rotate(L, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → L'_3
  91. Rotate(L'_1, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → A_1
  92. Segment(L'_3, A_1) → m_1
  93. Rotate(q_1, If[15 < l ≤ 16, ((-(l - 15)) * α / 2), ((-α)) / 2], D) → t_1
  94. Rotate(p_1, If[15 < l ≤ 16, ((-(l - 15)) * α / 2), ((-α)) / 2], B) → c_2
  95. Rotate(s_1, If[16 < l ≤ 17, ((-(l - 16)) * β / 2), ((-β)) / 2], A) → d_2
  96. Rotate(r_1, If[16 < l ≤ 17, ((-(l - 16)) * β / 2), ((-β)) / 2], E) → e_2
  97. Rotate(c_1, If[17 < l ≤ 18, ((l - 17) * π / 2), 1.5707963267948966], L) → f_2
  98. Intersect(g_2, m, 1) → D_1
  99. Segment(L, D_1) → h_2
  100. Dilate(L, If[20 < l ≤ 21, 21 - l, 0], D_1) → L'_4
  101. Segment(L, L'_4) → i_2
  102. Line(L, m) → j_2
  103. Point(j_2) → F_1
  104. Angle(D, L, F_1) → γ
  105. Dilate(L, If[21 < l ≤ 22, 22 - l, 0], D_1) → L'_5
  106. Vector(L, L'_5) → u_1
  107. Translate(f_2, u_1) → k_2
  108. Rotate(k_2, If[22 < l ≤ 23, ((l - 22) * γ), γ], D_1) → p_2
  109. Rotate(D_1, If[23 < l ≤ 24, (((l - 23) * 2) * π), (2 * π)], L) → G_1
  110. CircleArc(L, D_1, G_1) → q_2
  111. Angle(D_1, L, E_1) → δ
  112. Circle(D_1, (0.2 * o)) → r_2
  113. Intersect(r_2, m) → J_1, K_1
  114. Rotate(B, 45°, D_1) → B'_3
  115. Angle(B, D_1, B'_3) → ε
  116. Ray(D_1, B'_3) → l_2
  117. Intersect(m', l_2) → M_1
  118. Intersect(r_2, l_2, 1) → N_1
  119. Dilate(D_1, If[24 < l ≤ 25, 25 - l, 0], M_1) → L_1
  120. Segment(D_1, L_1) → m_2
  121. Intersect(r_2, i_2) → O_1,
  122. CircleSector(D_1, N_1, O_1) → s_2
  123. Dilate(C_1, If[25 < l ≤ 26, 26 - l, 0], Q_1) → P_1
  124. Segment(C_1, P_1) → n_2
  125. Dilate(Q_1, If[26 < l ≤ 27, 27 - l, 0], I_1) → R_1
  126. Segment(Q_1, R_1) → b_2
  127. Dilate(E_1, If[27 < l ≤ 28, 28 - l, 0], Q_1) → S_1
  128. Segment(E_1, S_1) → f_3
  129. Dilate(Q_1, If[28 < l ≤ 29, 29 - l, 0], H_1) → T_1
  130. Segment(Q_1, T_1) → g_3
  131. Rotate(s_2, If[30.5 < l ≤ 31.5, ((-(l - 30.5)) * π / 4), ((-π)) / 4], D_1) → c_3
  132. Line(Q_1, m_2) → h_3
  133. Point(h_3) → U_1
  134. Angle(I_1, Q_1, U_1) → θ
  135. Dilate(D_1, If[32 < l ≤ 33, 33 - l, 0], Q_1) → V_1
  136. Vector(D_1, V_1) → v_1
  137. Translate(c_3, v_1) → d_3
  138. Rotate(d_3, If[33 < l ≤ 34, ((-(l - 33)) * θ), (-θ)], Q_1) → e_3

AI 绘图提示语参考

绘制一个菱形ABCD，其中点A、B、C、D构成菱形的四个顶点。以点A为圆心，AB为半径画圆，与菱形的对角线交于点E。以点D为圆心，DA为半径画圆，与点B的圆交于点F。通过旋转和缩放操作，构造菱形的内切圆，并与菱形的边和对角线相交，生成一系列辅助线段和点，最终形成菱形的内切圆与菱形边的交点，以及菱形对角线的交点。
以上提示语为根据原始绘图命令整理而来，仅供参考。