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三角形面积证明（与矩形相关）

三角形面积证明（与矩形相关）
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2017-06-12 19:59
测试改名0函数解析几何
这个范例演示了如何通过构造矩形来证明三角形的面积公式。核心数学概念包括三角形面积公式、矩形面积公式以及几何变换。通过使用GeoGebra命令如Line、Point、Polygon等，可以直观地展示三角形与矩形之间的关系。适合初中及以上学生学习几何证明和面积计算。

GeoGebra 命令及参数

  1. Line(A, xAxis) → f_1
  2. Point(f_1) → M
  3. Ray(A, M) → h_1
  4. Point(h_1) → C
  5. Vector(C, A) → v
  6. Midpoint(B, C) → M_{BC}
  7. Rotate(A, (-ζ), M_{BC}) → I
  8. Segment(C, A) → i
  9. OrthogonalLine(I, i) → q
  10. Intersect(q, i) → G_1
  11. Vector(A, C) → u
  12. Midpoint(B, A) → M_{AB}
  13. Rotate(C, ε, M_{AB}) → J
  14. OrthogonalLine(J, i) → k
  15. Intersect(k, i) → E_1
  16. Polygon(A, B, C) → Area, c, a, b
  17. Angle(C, A, B) → α
  18. Angle(A, B, C) → β
  19. Angle(B, C, A) → γ
  20. Segment(A, B) → g
  21. Segment(B, C) → h
  22. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  23. OrthogonalLine(B, i) → j
  24. Intersect(j, i) → D
  25. Polygon(B, D, C) → poly2, c_2, b_2, d_1
  26. Polygon(B, A, D) → poly3, d_2, b_3, a_2
  27. Rotate(B, (-δ), M_{BC}) → B'
  28. Rotate(D, (-δ), M_{BC}) → L
  29. Rotate(C, (-δ), M_{BC}) → C'
  30. Polygon(B', L, C') → poly2', c', b', d'
  31. Rotate(B, δ, M_{AB}) → B'_1
  32. Rotate(A, δ, M_{AB}) → A'
  33. Rotate(D, δ, M_{AB}) → K
  34. Polygon(B'_1, A', K) → poly3', d'_1, b'_1, a'
  35. Segment(B, D) → d
  36. Rotate(A, ε, M_{AB}) → A'_1
  37. Rotate(B, ε, M_{AB}) → B'_2
  38. Polygon(A'_1, B'_2, J) → poly1', c'_1, a'_1, b'_2
  39. Polygon(Translate[J, Vector[(l * u)]], Translate[E_1, Vector[(l * u)]], Translate[A, Vector[(l * u)]]) → poly5, m, n, p
  40. Polygon(J, E_1, A) → poly4, a_3, c'_2, e_1
  41. Intersect(m, n) → F_1
  42. Rotate(B, (-ζ), M_{BC}) → B'_3
  43. Rotate(C, (-ζ), M_{BC}) → C'_2
  44. Polygon(I, B'_3, C'_2) → poly1'_1, c'_3, a'_2, b'_3
  45. Polygon(Translate[I, Vector[(o * v)]], Translate[G_1, Vector[(o * v)]], Translate[C, Vector[(o * v)]]) → poly7, r, s, t
  46. Polygon(I, G_1, C) → poly6, c_3, a'_3, g_1
  47. Intersect(r, s) → H_1
  48. Polygon(J, A, E_1) → poly8, e_2, c'_4, a_4
  49. Polygon(I, G_1, C) → poly9, c_4, a'_4, g_2
  50. Intersect(e_2, a_4) → I_1
  51. Intersect(e_2, c'_4) → J_1
  52. Point(a_4) → K_1
  53. Polygon(I_1, J_1, K_1) → poly10, k_1, i_1, j_1
  54. Mirror(C, A) → C'_1
  55. Mirror(A, C) → A'_2
  56. OrthogonalLine(C, b'_2) → l_1
  57. Intersect(l_1, b'_2) → E
  58. Polygon(E, C, B) → poly11, b_4, n_1, c_5
  59. Vector(B, J) → w
  60. Translate(E, w) → N'
  61. Translate(C, w) → C'_3
  62. Translate(B, w) → B'_4
  63. Polygon(Translate[E, Vector[(l * w)]], Translate[C, Vector[(l * w)]], Translate[B, Vector[(l * w)]]) → poly11', m_1, p_1, q_1
  64. Intersect(q_1, m_1) → F
  65. OrthogonalLine(A, b'_3) → r_1
  66. Intersect(b'_3, r_1) → G
  67. Polygon(B, G, A) → poly12, s_1, t_1, f_2
  68. Vector(B, I) → u_1
  69. Translate(B, u_1) → B'_5
  70. Translate(G, u_1) → H
  71. Translate(A, u_1) → A'_3
  72. Polygon(Translate[B, Vector[(o * u_1)]], Translate[G, Vector[(o * u_1)]], Translate[A, Vector[(o * u_1)]]) → poly12', h_2, i_2, j_2
  73. Intersect(h_2, i_2) → N