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几何作图范例：三角形与中心点

几何作图范例：三角形与中心点
几何画板及绘图内容加载中，请稍等……
2019-04-08 20:54
测试改名0圆角几何作图
①本范例演示了如何通过 GeoGebra 命令绘制三角形并找出其垂心、外心和重心；②涉及的核心数学概念包括三角形、角度、相似三角形、等边三角形、最小公倍数与最大公约数；③作图要点提炼自构造摘要，使用了 Polygon、OrthogonalLine、Intersect、Midpoint、Segment、Distance、LaTeX、Line、Angle、Rotate、Centroid、InteriorAngles、Text 等命令和对象；④适合高中阶段学生学习和理解几何作图技巧及三角形中心点的性质。

GeoGebra 命令及参数

  1. Polygon(A, B, C) → t1, c, a, b
  2. OrthogonalLine(A, a) → f
  3. OrthogonalLine(C, c) → g
  4. OrthogonalLine(B, b) → h
  5. Intersect(f, h) → Orthocenter
  6. Midpoint(c) → D
  7. Midpoint(b) → E
  8. Midpoint(a) → F
  9. OrthogonalLine(F, a) → i
  10. OrthogonalLine(E, b) → j
  11. OrthogonalLine(D, c) → k
  12. Intersect(j, i) → Circumcenter
  13. Segment(A, F) → l
  14. Segment(B, E) → m
  15. Segment(C, D) → n
  16. Intersect(l, n) → Centroid
  17. Segment(Orthocenter, Circumcenter) → p
  18. Distance(Orthocenter, Centroid) → distanceOrthocenterCentroid
  19. Distance(Centroid, Circumcenter) → distanceCentroidCircumcenter
  20. LaTeX(d, true, true) → text1
  21. Polygon(G, H, I) → t2, i_1, g_1, h_1
  22. Midpoint(h_1) → J
  23. Midpoint(i_1) → K
  24. Midpoint(g_1) → L
  25. Polygon(J, K, L) → t3, l_1, j_1, k_1
  26. Polygon(M, N, O) → t4, o, m_1, n_1
  27. Distance(M, P) → distanceMP
  28. Distance(P, N) → distancePN
  29. Distance(P, O) → distancePO
  30. LaTeX(e, true, true) → text2
  31. LaTeX(e, true, true) → text3
  32. Midpoint(n_1) → Q
  33. Midpoint(o) → R
  34. Midpoint(m_1) → S
  35. Polygon(T, U, V) → t5, v, t, u
  36. Polygon(T, U, 3) → poly1, q, r, s, W
  37. Polygon(V, T, 3) → poly2, f_1, p_1, q_1, Z
  38. Segment(Z, U) → r_1
  39. Segment(W, V) → s_1
  40. LaTeX(text5, true, true) → text6
  41. Polygon(A_1, B_1, C_1) → t6, c_1, a_1, b_1
  42. OrthogonalLine(B_1, b_1) → t_1
  43. OrthogonalLine(C_1, c_1) → d_1
  44. OrthogonalLine(A_1, a_1) → e_1
  45. Intersect(a_1, e_1) → D_1
  46. Intersect(t_1, b_1) → E_1
  47. Intersect(c_1, d_1) → F_1
  48. Polygon(F_1, D_1, E_1) → t7, f_2, g_2, h_2
  49. Polygon(G_1, H_1, I_1) → t8, i_2, j_2, k_2
  50. OrthogonalLine(J_1, k_2) → l_2
  51. OrthogonalLine(J_1, j_2) → m_2
  52. OrthogonalLine(J_1, i_2) → n_2
  53. Intersect(l_2, k_2) → K_1
  54. Intersect(n_2, i_2) → L_1
  55. Intersect(m_2, j_2) → M_1
  56. Polygon(L_1, M_1, K_1) → t9, p_2, q_2, r_2
  57. Polygon(N_1, O_1, P_1) → t10, s_2, t_2, o_1
  58. Line(N_1, Q_1) → a_2
  59. Line(P_1, Q_1) → b_2
  60. Line(O_1, Q_1) → c_2
  61. Intersect(b_2, s_2) → R_1
  62. Intersect(a_2, t_2) → S_1
  63. Intersect(c_2, o_1) → T_1
  64. Polygon(R_1, T_1, S_1) → t11, d_2, e_2, f_3
  65. Polygon(U_1, V_1, W_1) → t12, w_1, u_1, v_1
  66. Angle(U_1, V_1, W_1) → α
  67. Rotate(U_1, (1 / 3 * α), V_1) → A_2
  68. Rotate(W_1, (-((1 / 3 * α))), V_1) → Z_1
  69. Angle(W_1, U_1, V_1) → β
  70. Rotate(W_1, (1 / 3 * β), U_1) → B_{2}
  71. Rotate(V_1, (-1 / 3 * β), U_1) → C_2
  72. Angle(V_1, W_1, U_1) → γ
  73. Polygon(D_2, E_2, F_2) → t13, g_3, h_3, i_3
  74. Polygon(D_2, E_2, 3) → poly3, j_3, k_3, l_3, G_2
  75. Polygon(E_2, F_2, 3) → poly4, m_3, n_3, p_3, H_2
  76. Polygon(F_2, D_2, 3) → poly5, q_3, r_3, s_3, I_2
  77. Centroid(F_2, D_2, I_2) → J_2
  78. Centroid(H_2, E_2, F_2) → K_2
  79. Centroid(E_2, D_2, G_2) → L_2
  80. Segment(L_2, F_2) → t_3
  81. Segment(E_2, J_2) → b_3
  82. Segment(K_2, D_2) → a_3
  83. Intersect(b_3, a_3) → M_2
  84. InteriorAngles(t13) → ε, ζ, η
  85. Text("The napoleon point will be within the original triangle when all angles are less than 150 degrees.") → text7
  86. Text("when an angle equals 150 degrees, then the napoleon point will equal that vertex.") → text8