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正方形内的角度关系

正方形内的角度关系
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2017-03-19 11:44
圆多边形角变换几何作图
该范例展示了正方形内部涉及三个四分之一圆和若干线段的角度关系。核心数学概念包括角度和圆。通过使用GeoGebra命令如Polygon, Segment, Rotate, Circle等进行构造。适合初中及以上学生学习几何作图和角度关系。

GeoGebra 命令及参数

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Segment(D, A) → j
  3. Rotate(C, ((-π)) / 3, B) → C'_2
  4. Segment(B, C'_2) → l_2
  5. Circle(C'_2, (0.1 * j)) → d_1
  6. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A'_1
  7. Rotate(A'_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 3), ((-π)) / 3], B) → P
  8. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B) → A'
  9. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A''
  10. Rotate(A'', If[13 < e ≤ 14, ((-(e - 13)) * π / 3), ((-π)) / 3], B) → A'''_1
  11. Segment(P, A'''_1) → f''_1
  12. Rotate(A'', If[14 < e ≤ 15, ((e - 14) * π / 3), π / 3], C) → A'''_2
  13. Intersect(d_1, l_2, 1) → F_1
  14. Rotate(F_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 3), ((-π)) / 3], A'''_2) → G_1
  15. Midpoint(B, D) → H_1
  16. Midpoint(H_1, A'''_2) → R_1
  17. Circle(A, B) → e_1
  18. Segment(A, A') → f
  19. Circle(A, Radius[e_1] - f / 6) → p_1
  20. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  21. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  22. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  23. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  24. Segment(R', S) → f'''
  25. Point(f''') → Z
  26. Circle(Z, f / 6) → k_1
  27. OrthogonalLine(Z, f''') → g_1
  28. Intersect(k_1, g_1, 1) → A_1
  29. Point(f''') → W
  30. Circle(W, f / 6) → h_1
  31. OrthogonalLine(W, f''') → f_1
  32. Intersect(h_1, f_1, 1) → B_1
  33. Line(A_1, B_1) → j_1
  34. Segment(A'_1, A'') → f'
  35. Segment(A, B) → g
  36. Segment(B, C) → h
  37. Segment(C, D) → i
  38. Circle(B, (0.1 * j)) → p
  39. Circle(C, (0.15 * j)) → k
  40. Circle(D, (0.15 * j)) → r
  41. Segment(A, C) → l
  42. Segment(D, B) → m
  43. Intersect(l, m) → E
  44. Intersect(r, j, 1) → F
  45. Intersect(r, m, 1) → G
  46. Intersect(r, i, 1) → H
  47. Intersect(k, i, 1) → I
  48. Intersect(k, l, 1) → J
  49. Intersect(k, h, 1) → K
  50. Intersect(p, h, 1) → L
  51. Intersect(p, m, 1) → M
  52. Intersect(p, g, 1) → N
  53. Segment(R, A''') → f''
  54. CircleSector(B, L, N) → s
  55. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  56. Vector(B, B') → u
  57. Translate(s, u) → s'
  58. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  59. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  60. Vector(C, C') → v
  61. Translate(s'', v) → s'''
  62. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  63. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  64. Vector(D, D') → w
  65. Translate(t, w) → t'
  66. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  67. Dilate(A, If[10 < e ≤ 11, 11 - e, 0], E) → A'_2
  68. Dilate(E, If[11 < e ≤ 12, 12 - e, 0], C) → E'
  69. Dilate(D, If[12 < e ≤ 13, 13 - e, 0], E) → D'_1
  70. Dilate(E, If[13 < e ≤ 14, 14 - e, 0], B) → E'_1
  71. Dilate(D, If[17 < e ≤ 18, 18 - e, 0], E) → D'_2
  72. Vector(D, D'_2) → u_1
  73. Rotate(E, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5707963267948966], E) → E'_2
  74. Rotate(E'_1, If[14 < e ≤ 15, ((e - 14) * π / 2), 1.5708], E) → T
  75. Rotate(E'_2, If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5707963267948966], E) → U
  76. Rotate(T, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5708], E) → T'
  77. Rotate(U, If[17 < e ≤ 18, ((e - 17) * π / 2), 1.5707963267948966], E) → U'
  78. Rotate(T', If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5708], E) → T''
  79. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  80. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  81. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  82. Vector(B, B'_1) → v_1
  83. Vector(B, B'_3) → w_1
  84. Vector(B, B'_2) → i_1
  85. Circle(B, C) → c_1
  86. Intersect(e_1, j_1, 1) → C_1
  87. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_3
  88. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_4
  89. CircleArc(B, A'_4, A) → r_1
  90. Rotate(B, If[11 < e ≤ 12, ((e - 11) * π / 2), π / 2], A) → B'_4
  91. CircleArc(A, B, B'_4) → s_1
  92. Segment(A, B) → n_1
  93. Circle(A, 1.6411491042356954) → t_1
  94. Rotate(B, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], C) → B'_5
  95. CircleArc(C, B'_5, B) → q
  96. Intersect(q, r_1) → O, , ,
  97. Rotate(A'_1, If[14 < e ≤ 15, ((e - 14) * π / 3), π / 3], C) → Q
  98. Segment(Q, A'''_2) → f''_2
  99. Polygon(B, A'''_2, Q) → poly2, q_1, b_1, a'''_2
  100. Intersect(p, f') → V,
  101. Intersect(p, f''_1) → D_1,
  102. Rotate(V, If[15 < e ≤ 16, ((-(e - 15)) * π / 3), ((-π)) / 3], B) → V'
  103. Intersect(d_1, f''_2, 1) → E_1
  104. Dilate(B, If[16 < e ≤ 17, 17 - e, 0], H_1) → B'_6
  105. Segment(B, B'_6) → n
  106. Rotate(B, If[17 < e ≤ 18, ((e - 17) * π), π], H_1) → B'_7
  107. Rotate(B'_6, If[17 < e ≤ 18, ((e - 17) * π), π], H_1) → I_1
  108. Segment(B'_7, I_1) → n'
  109. Intersect(r_1, s_1, 1) → J_1
  110. Segment(B, J_1) → l_1
  111. Segment(Q, O) → m_1
  112. Intersect(l_1, m_1) → K_1
  113. Segment(K_1, H_1) → a_1
  114. Line(A'''_2, a_1) → f_2
  115. Line(A'''_2, n) → g_2
  116. Point(g_2) → L_1
  117. Point(f_2) → M_1
  118. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], J_1) → B'_8
  119. Segment(B, B'_8) → h_2
  120. Dilate(C, If[19 < e ≤ 20, 20 - e, 0], K_1) → C'_1
  121. Segment(Q, C'_1) → i_2
  122. Dilate(K_1, If[20 < e ≤ 21, 21 - e, 0], O) → N_1
  123. Segment(K_1, N_1) → j_2
  124. Dilate(K_1, If[21 < e ≤ 22, 22 - e, 0], H_1) → O_1
  125. Segment(K_1, O_1) → k_2
  126. Dilate(A'''_2, If[22 < e ≤ 23, 23 - e, 0], H_1) → P_1
  127. Vector(A'''_2, P_1) → u_2
  128. CircleSector(C'_2, G_1, F_1) → c_2
  129. Translate(c_2, u_2) → d_2
  130. Angle(B, A'''_2, M_1) → α
  131. Rotate(d_2, If[23 < e ≤ 24, (((e - 23) * 5) * π / 4), (5 * π / 4)], H_1) → e_2
  132. Dilate(A'''_2, If[24 < e ≤ 25, 25 - e, 0], R_1) → Q_1
  133. Vector(A'''_2, Q_1) → v_2
  134. Translate(c_2, v_2) → p_2
  135. Rotate(p_2, If[25 < e ≤ 26, ((e - 25) * ((5 * π / 4) - π / 3)), (5 * π / 4) - π / 3], R_1) → q_2
  136. Dilate(R_1, If[26 < e ≤ 27, 27 - e, 0], H_1) → S_1
  137. Vector(R_1, S_1) → w_2
  138. Translate(q_2, w_2) → r_2
  139. Segment(C'_2, C) → m_2