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维维安定理演示

维维安定理演示
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2019-03-27 04:04
测试改名0三角函数几何作图
①本范例演示维维安定理在三角形内的应用，即三角形的重心到三边的距离之和等于该三角形高的三分之一；②涉及核心数学概念包括三角形重心、高、距离、维维安定理；③作图要点：使用Circle（圆）、Point（点）、Translate（平移）、Polygon（多边形）、OrthogonalLine（垂线）、Intersect（交点）、Line（直线）、Segment（线段）、Centroid（重心）、Max（最大值）、Rotate（旋转）、Midpoint（中点）、Vector（向量）等命令和对象构建三角形、高、重心及相关线段；④适合高中及以上数学学习者和教师用于理解和验证维维安定理。

GeoGebra 命令及参数

  1. Circle(V, W) → c_1
  2. Point(c_1, b_1 / 8) → D
  3. Translate(D, Vector[(c * Vector[(4, 0)])]) → D'
  4. Translate(A, Vector[(c * Vector[(4, 0)])]) → A'
  5. Point(xAxis) → B
  6. Translate(B, Vector[(c * Vector[(4, 0)])]) → B'
  7. Polygon(A, B, 3) → poly1, f, g, h, C
  8. OrthogonalLine(D, Segment[A, C]) → j
  9. Intersect(j, Segment[A, C]) → F
  10. Translate(F, Vector[(c * Vector[(4, 0)])]) → F'
  11. OrthogonalLine(D, Segment[B, C]) → i
  12. Intersect(i, Segment[B, C]) → E
  13. Translate(E, Vector[(c * Vector[(4, 0)])]) → E'
  14. Line(D', Rotate[E', 30°, D']) → p
  15. Line(D', Rotate[E', (-(30°)), D']) → q
  16. Translate(C, Vector[(c * Vector[(4, 0)])]) → C'_1
  17. Segment(A', C'_1) → s
  18. Intersect(s, Line[D', Rotate[F', (-(30°)), D']]) → K
  19. Intersect(q, s) → P
  20. Translate(C, Vector[(c * Vector[(4, 0)])]) → C'
  21. Segment(B', C') → r
  22. Intersect(p, r) → H
  23. Intersect(q, r) → O
  24. Centroid(Polygon[O, P, C'_1]) → N
  25. Intersect(q, s) → J
  26. Polygon(K, D', J) → t1, j_2, k_2, d'
  27. Centroid(t1) → L
  28. Max(Min[a, 1], 0) → b
  29. Segment(D, ((1 - b) * D) + (b * F)) → n
  30. Translate(poly1, Vector[(c * Vector[(4, 0)])]) → poly1'
  31. OrthogonalLine(D, Segment[A, B]) → k
  32. Intersect(k, Segment[A, B]) → G
  33. Segment(D, ((1 - b) * D) + (b * G)) → l
  34. Segment(D, ((1 - b) * D) + (b * E)) → m
  35. Translate(m, Vector[(c * Vector[(4, 0)])]) → m'
  36. Translate(G, Vector[(c * Vector[(4, 0)])]) → G'
  37. Translate(A, Vector[(c * Vector[(4, 0)])]) → A'_1
  38. Translate(B, Vector[(c * Vector[(4, 0)])]) → B'_1
  39. Segment(A'_1, B'_1) → t
  40. Intersect(q, r) → I
  41. Segment(K, ((1 - d) * K) + (d * J)) → j_1
  42. Segment(H, ((1 - d) * H) + (d * I)) → k_1
  43. Polygon(H, I, D') → t2, l_1, h_2, i_2
  44. Centroid(t2) → M
  45. Rotate(Rotate[K, ((-e) * 120°), L], ((-o) * 120°), N) → K'
  46. Rotate(Rotate[J, ((-e) * 120°), L], ((-o) * 120°), N) → J'
  47. Rotate(Rotate[D', ((-e) * 120°), L], ((-o) * 120°), N) → D''
  48. Translate(Rotate[H, ((-o) * 120°), N], Vector[((0.5 * u) * Vector[K', H])]) → H'
  49. Translate(Rotate[I, ((-o) * 120°), N], Vector[((0.5 * u) * Vector[K', H])]) → I'
  50. Translate(Rotate[D', ((-o) * 120°), N], Vector[((0.5 * u) * Vector[K', H])]) → D''_1
  51. Intersect(p, t) → Q
  52. Line(D', Rotate[G', 30°, D']) → m_1
  53. Intersect(m_1, t) → R
  54. Segment(R, ((1 - d) * R) + (d * Q)) → q_1
  55. Midpoint(Segment[A, B]) → S
  56. Midpoint(Segment[C, B]) → T
  57. Midpoint(Segment[C, A]) → U
  58. Polygon(D', R, Q) → t3, q_2, r_1, r_2
  59. Translate(D', Vector[(u * Vector[G, S])]) → D''_2
  60. Translate(R, Vector[(u * Vector[G, S])]) → R'
  61. Translate(Q, Vector[(u * Vector[G, S])]) → Q'
  62. Segment(A, B) → s_1
  63. Segment(C, B) → t_1
  64. Segment(C, A) → a_1
  65. Line(D', t) → f_1
  66. Intersect(s, f_1) → Z
  67. Intersect(r, k_1) → A_1
  68. Intersect(r, k_1) → B_1
  69. Intersect(f_1, r) → C_1
  70. Rotate(Polygon[Z, C_1, ((1 - e) * C') + (e * Mirror[C', Segment[Z, C_1]])], ((-u) * 60°), C_1) → t4
  71. Vector((x(C) - x(D), 0)) → v
  72. Segment(Z, ((1 - d) * Z) + (d * C_1)) → g_1
  73. Rotate(Segment[Translate[D', Vector[(o * Vector[F', Z])]], Translate[D_1, Vector[(o * Vector[F', Z])]]], ((-u) * 60°), C_1) → m'_1
  74. Rotate(Translate[n, Vector[(c * Vector[(4, 0)]) + (o * Vector[F', Z])]], ((-u) * 60°), C_1) → n'
  75. Translate(l, Vector[(c * Vector[(4, 0)]) + (w * v)]) → l'