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等边三角形的特殊性质

等边三角形的特殊性质
几何画板及绘图内容加载中，请稍等……
2016-07-25 10:54
测试改名0角向量解析几何几何作图
所属专题 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2
这个范例演示了等边三角形的特殊性质。核心数学概念包括等边三角形的性质、中心、正交线、圆和角度。通过使用GeoGebra命令如Polygon、Segment、Centroid、OrthogonalLine、Circle、Angle等，可以构造等边三角形并探索其性质。适合初中及以上学生学习几何作图和等边三角形的相关概念。

GeoGebra 命令及参数

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. Segment(A, B) → i
  3. Segment(B, C) → j
  4. Segment(C, A) → k
  5. Centroid(poly1) → D
  6. OrthogonalLine(D, g) → l
  7. OrthogonalLine(D, i) → m
  8. Intersect(l, g) → E
  9. Circle(D, E) → c
  10. Angle(B, A, C) → α
  11. Rotate(B, 30°, A) → B'
  12. Angle(B, A, B') → β
  13. Line(A, B') → n
  14. Intersect(c, n, 1) → F
  15. Midpoint(A, F) → G
  16. Tangent(F, c) → p,
  17. Intersect(p, i) → H
  18. Intersect(p, h) → I
  19. Polygon(A, H, I) → poly2, i_1, a, h_1
  20. Centroid(poly2) → J
  21. Circle(J, F) → d
  22. Intersect(c, m, 2) → K
  23. Tangent(K, c) → q,
  24. Intersect(q, g) → L
  25. Intersect(q, h) → M
  26. Polygon(M, L, C) → poly3, c_1, m_1, l_1
  27. Centroid(poly3) → N
  28. Circle(N, K) → e
  29. Line(B, D) → r
  30. Intersect(h, k) → O
  31. Intersect(r, h) → P
  32. Intersect(c, r) → Q, R
  33. Tangent(Q, c) → s,
  34. Intersect(s, g) → S
  35. Intersect(s, i) → T
  36. Polygon(T, B, S) → poly4, s_1, t, b
  37. Centroid(poly4) → U
  38. Circle(U, Q) → d_1
  39. Point(cc) → V
  40. Tangent(V, ee) → f_1, g_1
  41. Tangent(V, dd) → j_1, k_1
  42. Tangent(V, ddd) → n_1, p_1
  43. Intersect(e, ee, 1) → W
  44. Intersect(e, ee) → Z, , ,
  45. Intersect(ee, g_1) → A_1
  46. Intersect(dd, j_1) → B_1
  47. Intersect(ddd, n_1) → C_1
  48. Segment(V, A_1) → q_1
  49. Segment(V, B_1) → r_1
  50. Segment(V, C_1) → t_1
  51. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
  52. Segment(A, A') → a_1
  53. Rotate(A, If[1 < o ≤ 2, ((o - 1) * π / 3), π / 3], A) → A'_1
  54. Rotate(A', If[1 < o ≤ 2, ((o - 1) * π / 3), π / 3], A) → A''
  55. Segment(A'_1, A'') → b_1
  56. Rotate(A, If[2 < o ≤ 3, ((-(o - 2)) * π / 3), ((-π)) / 3], B) → A'_2
  57. Rotate(A', If[2 < o ≤ 3, ((-(o - 2)) * π / 3), ((-π)) / 3], B) → A''_1
  58. Segment(A'_2, A''_1) → e_1
  59. Max(list1) → u
  60. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], A_1) → V'
  61. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], C_1) → V'_2
  62. Circle(V, q_1) → c_2
  63. Intersect(c_2, c, 1) → D_1
  64. Intersect(c_2, c) → E_1, F_1, ,
  65. Intersect(c_2, r_1) → G_1,
  66. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], G_1) → V'_1
  67. Dilate(G_1, If[5 < o ≤ 6, 6 - o, 0], B_1) → H_1
  68. Segment(V, V') → f_2
  69. Segment(V, V'_1) → g_2
  70. Segment(V, V'_2) → h_2
  71. Segment(G_1, H_1) → i_2
  72. Angle(A_1, V, B_1) → γ
  73. Rotate(V, If[6 < o ≤ 7, ((o - 6) * γ), γ], V) → V'_3
  74. Rotate(V', If[6 < o ≤ 7, ((o - 6) * γ), γ], V) → V''
  75. Segment(V'_3, V'') → j_2
  76. Dilate(V, If[7 < o ≤ 8, 8 - o, 0], G_1) → V'_4
  77. Vector(V, V'_4) → v
  78. Translate(V, v) → V'_5
  79. Translate(V'_2, v) → I_1
  80. Segment(V'_5, I_1) → k_2
  81. Angle(B_1, V, C_1) → δ
  82. Rotate(V'_5, If[8 < o ≤ 9, ((-(o - 8)) * δ), (-δ)], V'') → J_1
  83. Rotate(I_1, If[8 < o ≤ 9, ((-(o - 8)) * δ), (-δ)], V'') → K_1
  84. Segment(J_1, K_1) → l_2
  85. Circle(V, r_1) → d_2
  86. Intersect(d_2, t_1) → L_1,
  87. Segment(V, V') → m_2
  88. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], B_1) → V'_6
  89. Segment(V, V'_6) → n_2
  90. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], L_1) → V'_7
  91. Segment(V, V'_7) → p_2
  92. Dilate(L_1, If[5 < o ≤ 6, 6 - o, 0], C_1) → M_1
  93. Segment(L_1, M_1) → q_2
  94. Angle(B_1, V, L_1) → ε
  95. Rotate(V, If[6 < o ≤ 7, ((o - 6) * ε), ε], V) → V'_8
  96. Rotate(V'_6, If[6 < o ≤ 7, ((o - 6) * ε), ε], V) → N_1
  97. Segment(V'_8, N_1) → r_2
  98. Dilate(V, If[7 < o ≤ 8, 8 - o, 0], N_1) → V'_9
  99. Vector(V, V'_9) → w
  100. Translate(V, w) → V'_{10}
  101. Translate(V', w) → V''_1
  102. Segment(V'_{10}, V''_1) → s_2
  103. Angle(N_1, V, A_1) → ζ
  104. Rotate(V'_{10}, If[8 < o ≤ 9, ((-(o - 8)) * ζ), (-ζ)], N_1) → O_1
  105. Rotate(V''_1, If[8 < o ≤ 9, ((-(o - 8)) * ζ), (-ζ)], N_1) → P_1
  106. Segment(O_1, P_1) → t_2
  107. Circle(V, t_1) → e_2
  108. Intersect(e_2, q_1) → Q_1,
  109. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], Q_1) → V'_{11}
  110. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], B_1) → V'_{12}
  111. Dilate(V, If[4 < o ≤ 5, 5 - o, 0], C_1) → V'_{13}
  112. Segment(V, V'_{11}) → a_2
  113. Segment(V, V'_{12}) → b_2
  114. Segment(V, V'_{13}) → f_3
  115. Dilate(Q_1, If[5 < o ≤ 6, 6 - o, 0], A_1) → R_1
  116. Segment(Q_1, R_1) → g_3
  117. Angle(C_1, V, Q_1) → η
  118. Rotate(V, If[6 < o ≤ 7, ((o - 6) * η), η], V) → V'_{14}
  119. Rotate(V'_{13}, If[6 < o ≤ 7, ((o - 6) * η), η], V) → S_1
  120. Segment(V'_{14}, S_1) → h_3
  121. Dilate(V, If[7 < o ≤ 8, 8 - o, 0], S_1) → V'_{15}
  122. Vector(V, V'_{15}) → u_1
  123. Translate(V, u_1) → V'_{16}
  124. Translate(V'_{12}, u_1) → T_1
  125. Segment(V'_{16}, T_1) → i_3
  126. Angle(A_1, V, B_1) → θ
  127. Rotate(V'_{16}, If[8 < o ≤ 9, ((-(o - 8)) * θ), (-θ)], S_1) → U_1
  128. Rotate(T_1, If[8 < o ≤ 9, ((-(o - 8)) * θ), (-θ)], S_1) → V_1
  129. Segment(U_1, V_1) → j_3

AI 绘图提示语参考

画一个等边三角形，标出其重心，并画出从重心到三边的垂线。以重心为圆心，画一个经过三角形顶点的圆。测量三角形内角，并旋转一个顶点30度得到新的顶点，连接原顶点和旋转后的顶点。在圆上找到与垂线交点，并连接这些交点形成新的多边形。继续以这些交点为圆心画圆，并找到新的交点和切线，最终形成多个嵌套的多边形和圆。
以上提示语为根据原始绘图命令整理而来，仅供参考。