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余弦定理扩展勾股定理

余弦定理扩展勾股定理
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2017-06-06 04:43
测试改名0三角函数不等式几何作图
这个范例演示了余弦定理如何通过 -2abcosC 项在 a^2 + b^2 与 c^2 之间进行加或减以维持等式平衡。通过观察不同情况下的变化，可以深入理解余弦定理的应用。涉及的核心数学概念包括余弦定理、勾股定理、三角形和不等式。作图要点包括使用点、线、距离、圆弧、角度、线段、多边形、旋转、向量、平移、缩放、镜像、面积、求和、垂线、交点、射线等 GeoGebra 命令和对象。适合高中及以上数学学习者使用，用于理解和应用余弦定理和勾股定理。

GeoGebra 命令及参数

  1. Point(Circle[A, scale]) → arcright
  2. Line(A, arcright) → h
  3. Point(h) → C
  4. Distance(A, C) → ac
  5. Point(Circle[A, scale]) → arcleft
  6. CircleArc(A, arcright, arcleft) → c_1
  7. Point(c_1) → B
  8. Distance(B, C) → bc
  9. Angle(B, C, A) → angleC
  10. Segment(A, arcleft) → f
  11. Segment(A, arcright) → g
  12. Segment(C, A) → b
  13. Segment(A, B) → c
  14. Segment(B, C) → a
  15. Polygon(A, B, 4) → csquared, j, k, l, m, D, E
  16. Polygon(B, C, 4) → asquared, n, p, q, r, F, G
  17. Polygon(C, A, 4) → bsquared, s, t, a_1, d, H, I
  18. Angle(B, C, I) → α
  19. Angle(F, C, A) → β
  20. Rotate(C, (-β), C) → C'
  21. Rotate(A, (-β), C) → A'
  22. Segment(C', A') → b'_1
  23. Rotate(B, α, C) → B'
  24. Rotate(C, α, C) → C'_1
  25. Segment(B', C'_1) → a'_1
  26. Vector(C, I) → u
  27. Translate(B', u) → B''
  28. Translate(C'_1, u) → J
  29. Segment(B'', J) → a''
  30. Vector(C, F) → v
  31. Translate(C', v) → C''
  32. Translate(A', v) → A''
  33. Segment(C'', A'') → b''
  34. Dilate(B'', cosc, J) → B'''
  35. Dilate(J, cosc, J) → J'
  36. Segment(B''', J') → a'''
  37. Dilate(C'', cosc, C'') → C'''
  38. Dilate(A'', cosc, C'') → A'''
  39. Segment(C''', A''') → b'''
  40. Vector(J, H) → w
  41. Translate(B''', w) → K
  42. Translate(J', w) → J''
  43. Segment(K, J'') → i
  44. Vector(C'', G) → e
  45. Translate(C''', e) → L
  46. Translate(A''', e) → M
  47. Segment(L, M) → f_1
  48. Polygon(C'', A''', M, G) → poly1, c'', a'''_1, m_1, g_1
  49. Polygon(B''', K, H, J) → poly2, b'''_1, k_1, h_1, j_1
  50. Line(H, J) → i_1
  51. Line(G, C'') → l_1
  52. Mirror(C'', l_1) → C'''_1
  53. Mirror(A''', l_1) → N
  54. Mirror(M, l_1) → M'
  55. Mirror(G, l_1) → G'
  56. Polygon(C'''_1, N, M', G') → poly1', bcosc, a'_2, m', g'
  57. Mirror(B''', i_1) → O
  58. Mirror(K, i_1) → K'
  59. Mirror(H, i_1) → H'
  60. Mirror(J, i_1) → J'_1
  61. Polygon(O, K', H', J'_1) → poly2', b'_2, acosc, h', j'_1
  62. Area(csquared) → cc
  63. Area(asquared) → aa
  64. Area(bsquared) → bb
  65. Sum({aa, bb}) → aaplusbb
  66. Distance(A, B) → ab
  67. OrthogonalLine(B, b) → n_1
  68. OrthogonalLine(B, h) → p_1
  69. Intersect(h, p_1) → Q
  70. Point(t) → R
  71. Intersect(b'_2, acosc) → S
  72. Segment(R, S) → s_1
  73. Intersect(b'_2, acosc) → T
  74. Intersect(b'_2, j'_1) → U
  75. Segment(T, U) → t_1
  76. Segment(R, T) → acosC
  77. Segment(T, U) → b'
  78. Segment(C'''_1, N) → bcosC
  79. Segment(N, M') → a'
  80. Angle(A, B, C) → γ
  81. Ray(A, C) → q_1