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动态几何演示：涉及正方形、圆与切圆的定理

动态几何演示：涉及正方形、圆与切圆的定理
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2017-03-18 10:51
圆多边形向量几何作图
①本范例动态演示一个涉及正方形、两圆及另一相切圆的定理；②核心数学概念包括圆、正方形、几何均值、向量、几何变换；③构造要点：使用 Circle, Rotate, Dilate, Segment, Polygon, Point, OrthogonalLine, Intersect, Line, Radius, CircleSector, Vector, Translate, CircleArc, Midpoint 等命令绘制图形；④适合高中及以上数学学习者和教师用于理解几何定理和变换。

GeoGebra 命令及参数

  1. Circle(A, B) → e_1
  2. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A'_1
  3. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B) → A'
  4. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A''
  5. Segment(A, A') → f
  6. Circle(A, Radius[e_1] - f / 6) → p_1
  7. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  8. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  9. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  10. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  11. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  12. Segment(R', S) → f'''
  13. Point(f''') → Z
  14. Circle(Z, f / 6) → k_1
  15. OrthogonalLine(Z, f''') → g_1
  16. Intersect(k_1, g_1, 1) → A_1
  17. Point(f''') → W
  18. Circle(W, f / 6) → h_1
  19. OrthogonalLine(W, f''') → f_1
  20. Intersect(h_1, f_1, 1) → B_1
  21. Line(A_1, B_1) → j_1
  22. Intersect(p_1, j_1, 1) → V
  23. Circle(V, f / 6) → q_1
  24. Radius(q_1) → n
  25. Segment(A'_1, A'') → f'
  26. Segment(A, B) → g
  27. Segment(B, C) → h
  28. Segment(C, D) → i
  29. Segment(D, A) → j
  30. Circle(B, (0.35 * n)) → p
  31. Circle(C, (0.15 * j)) → k
  32. Circle(A, (0.15 * j)) → q
  33. Circle(D, (0.15 * j)) → r
  34. Segment(A, C) → l
  35. Segment(D, B) → m
  36. Intersect(l, m) → E
  37. Intersect(r, j, 1) → F
  38. Intersect(r, m, 1) → G
  39. Intersect(r, i, 1) → H
  40. Intersect(k, i, 1) → I
  41. Intersect(k, l, 1) → J
  42. Intersect(k, h, 1) → K
  43. Intersect(p, h, 1) → L
  44. Intersect(p, m, 1) → M
  45. Intersect(p, g, 1) → N
  46. Intersect(q, g, 1) → O
  47. Intersect(q, l, 1) → P
  48. Intersect(q, j, 1) → Q
  49. Segment(R, A''') → f''
  50. CircleSector(B, L, N) → s
  51. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  52. Vector(B, B') → u
  53. Translate(s, u) → s'
  54. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  55. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  56. Vector(C, C') → v
  57. Translate(s'', v) → s'''
  58. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  59. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  60. Vector(D, D') → w
  61. Translate(t, w) → t'
  62. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  63. Dilate(A, If[10 < e ≤ 11, 11 - e, 0], E) → A'_2
  64. Dilate(E, If[11 < e ≤ 12, 12 - e, 0], C) → E'
  65. Dilate(D, If[12 < e ≤ 13, 13 - e, 0], E) → D'_1
  66. Dilate(E, If[13 < e ≤ 14, 14 - e, 0], B) → E'_1
  67. Dilate(D, If[17 < e ≤ 18, 18 - e, 0], E) → D'_2
  68. Vector(D, D'_2) → u_1
  69. Rotate(E, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5707963267948966], E) → E'_2
  70. Rotate(E'_1, If[14 < e ≤ 15, ((e - 14) * π / 2), 1.5708], E) → T
  71. Rotate(E'_2, If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5707963267948966], E) → U
  72. Rotate(T, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5708], E) → T'
  73. Rotate(U, If[17 < e ≤ 18, ((e - 17) * π / 2), 1.5707963267948966], E) → U'
  74. Rotate(T', If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5708], E) → T''
  75. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  76. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  77. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  78. Vector(B, B'_1) → v_1
  79. Vector(B, B'_3) → w_1
  80. Vector(B, B'_2) → i_1
  81. Circle(B, C) → c_1
  82. Intersect(e_1, j_1, 1) → C_1
  83. Intersect(c_1, q_1) → D_1, E_1, ,
  84. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_3
  85. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_4
  86. CircleArc(B, A'_4, A) → r_1
  87. Rotate(B, If[11 < e ≤ 12, ((e - 11) * π / 2), π / 2], A) → B'_4
  88. CircleArc(A, B, B'_4) → s_1
  89. Rotate(D_1, If[12 < e ≤ 14, ((e - 12) * π), (2 * π)], V) → F_1
  90. CircleArc(V, D_1, F_1) → d_1
  91. OrthogonalLine(V, f''') → l_1
  92. Intersect(l_1, f''') → G_1
  93. Dilate(V, If[14 < e ≤ 15, 15 - e, 0], G_1) → V'
  94. Segment(V, V') → m_1
  95. Dilate(D, If[15 < e ≤ 16, 16 - e, 0], G_1) → D'_3
  96. Vector(D, D'_3) → a_1
  97. Translate(t, a_1) → t'_1
  98. Segment(A, B) → n_1
  99. Circle(A, 1.6411491042356954) → t_1
  100. Circle(A, m_1) → c_2
  101. Intersect(c_2, n_1, 1) → H_1
  102. Circle(H_1, m_1) → d_2
  103. Intersect(d_2, n_1, 2) → I_1
  104. Circle(I_1, m_1) → e_2
  105. Intersect(e_2, n_1, 2) → J_1
  106. Circle(J_1, m_1) → f_2
  107. Intersect(f_2, n_1, 2) → K_1
  108. Circle(K_1, m_1) → g_2
  109. Intersect(g_2, n_1, 2) → L_1
  110. Dilate(G_1, If[16 < e ≤ 17, 17 - e, 0], L_1) → M_1
  111. Vector(G_1, M_1) → b_1
  112. Translate(V, b_1) → V'_1
  113. Translate(V', b_1) → V''
  114. Segment(V'_1, V'') → h_2
  115. Dilate(G_1, If[17 < e ≤ 18, 18 - e, 0], K_1) → N_1
  116. Vector(G_1, N_1) → u_2
  117. Translate(V, u_2) → V'_2
  118. Translate(V', u_2) → V''_1
  119. Segment(V'_2, V''_1) → i_2
  120. Dilate(G_1, If[18 < e ≤ 19, 19 - e, 0], J_1) → P_1
  121. Vector(G_1, P_1) → v_2
  122. Translate(V, v_2) → V'_3
  123. Translate(V', v_2) → V''_2
  124. Segment(V'_3, V''_2) → j_2
  125. Dilate(G_1, If[19 < e ≤ 20, 20 - e, 0], I_1) → Q_1
  126. Vector(G_1, Q_1) → w_2
  127. Translate(V, w_2) → V'_4
  128. Translate(V', w_2) → V''_3
  129. Segment(V'_4, V''_3) → k_2
  130. Dilate(G_1, If[20 < e ≤ 21, 21 - e, 0], A) → R_1
  131. Vector(G_1, R_1) → a_2
  132. Translate(V, a_2) → V'_5
  133. Translate(V', a_2) → V''_4
  134. Segment(V'_5, V''_4) → l_2
  135. Midpoint(G_1, I_1) → O_1
  136. Dilate(G_1, If[21 < e ≤ 22, 22 - e, 0], O_1) → S_1
  137. Vector(G_1, S_1) → b_2
  138. Translate(V, b_2) → V'_6
  139. Translate(V', b_2) → V''_5
  140. Segment(V'_6, V''_5) → m_2
  141. Rotate(V'_6, If[22 < e ≤ 23, (((e - 22) * 3) * π / 4), (3 * π / 4)], V'_6) → T_1
  142. Rotate(V''_5, If[22 < e ≤ 23, (((e - 22) * 3) * π / 4), (3 * π / 4)], V'_6) → U_1
  143. Segment(T_1, U_1) → n_2
  144. Dilate(U_1, If[23 < e ≤ 24, 24 - e, 0], I_1) → V_1
  145. Vector(U_1, V_1) → p_2
  146. Translate(T_1, p_2) → W_1
  147. Translate(U_1, p_2) → Z_1
  148. Segment(W_1, Z_1) → q_2
  149. Rotate(W_1, If[24 < e ≤ 25, ((e - 24) * π / 4), π / 4], I_1) → A_2
  150. Rotate(Z_1, If[25 < e ≤ 26, ((e - 25) * π / 4), π / 4], I_1) → B_2
  151. Segment(A_2, B_2) → r_2