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Geometric Proof of the Cosine Law

Geometric Proof of the Cosine Law
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2018-10-29 03:45
PolygonTrigonometryGeometric Construction
This example demonstrates the geometric proof of the Cosine Law using GeoGebra. It involves core mathematical concepts such as the Cosine Law, properties of rhombuses, and kites. The construction utilizes GeoGebra commands and objects like Intersect, CircleArc, Point, Translate, Polygon, Segment, Length, Angle, Rotate, and Ray. Suitable for high school students learning trigonometry and geometric proofs, helping them understand the derivation and application of the Cosine Law.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. CircleArc((0, 0), (4, 0), (0, 4)) → c_1
  3. Point(c_1) → B
  4. Translate(B, (8, 0)) → J
  5. Point(Circle[A, 3]) → E
  6. Polygon(A, E, 4) → poly2, l, m, n, p, F, G
  7. Segment(G, B) → q
  8. Length(q) → cLength
  9. Segment(A, B) → f
  10. Polygon(A, B, 4) → poly1, g, h, i, j, C, D
  11. Segment(A, E) → k
  12. Angle(G, A, B) → α
  13. Rotate(A, (-α), D) → A'
  14. Angle(A', D, A) → β
  15. Ray(D, A') → r
  16. Rotate(A, α, E) → A'_1
  17. Angle(A, E, A'_1) → γ
  18. Ray(E, A'_1) → s
  19. Intersect(r, s) → H
  20. Segment(D, H) → t
  21. Segment(H, A) → a
  22. Segment(H, E) → b
  23. Translate(G, (8, 0)) → I
  24. Segment(I, J) → c
  25. Polygon(I, J, 4) → poly3, d, e, f_1, g_1, K, L
  26. Angle(A, B, G) → δ
  27. Angle(B, G, A) → ε
  28. Rotate(K, (-δ), J) → K'
  29. Angle(K', J, K) → ζ
  30. Ray(J, K') → h_1
  31. Rotate(J, ε, K) → J'
  32. Angle(J, K, J') → η
  33. Ray(K, J') → i_1
  34. Intersect(h_1, i_1) → M
  35. Rotate(L, (-δ), K) → L'
  36. Angle(L', K, L) → θ
  37. Ray(K, L') → j_1
  38. Rotate(K, ε, L) → K'_1
  39. Angle(K, L, K'_1) → ι
  40. Ray(L, K'_1) → k_1
  41. Intersect(j_1, k_1) → N
  42. Rotate(I, (-δ), L) → I'
  43. Angle(I', L, I) → κ
  44. Ray(L, I') → l_1
  45. Rotate(L, ε, I) → L'_1
  46. Angle(L, I, L'_1) → λ
  47. Ray(I, L'_1) → m_1
  48. Intersect(l_1, m_1) → O
  49. Segment(J, M) → n_1
  50. Segment(M, K) → p_1
  51. Segment(K, N) → q_1
  52. Segment(N, L) → r_1
  53. Segment(L, O) → s_1
  54. Segment(O, I) → t_1
  55. Translate(E, (8, 0)) → E'
  56. Translate(D, (8, 0)) → D'
  57. Polygon(M, D', N, K) → parallel1, m_2, d', n_2, k_2
  58. Polygon(N, L, O, E') → parallel2, n_3, l_2, o, e'
  59. Point(c_1) → P
  60. Polygon((8, -3), (8, 1), (11, 1), (11, -3)) → finalQuad, a_1, b_1, d_1, e_1
  61. Polygon((8, 1), (8, 4), (12, 4), (12, 1)) → finalQuad1, f_2, g_2, h_2, i_2
  62. Polygon(D, A, H) → t1, h_3, d_2, a_2
  63. Intersect(h_3, d_2) → Q
  64. Intersect(d_2, a_2) → R
  65. Polygon(E, Q, R) → t2, r_2, e_2, q_2
  66. Polygon(Q, B, G) → t3, g_3, q_3, b_2
  67. Point(n_3) → S
  68. Intersect(n_2, n_3) → T
  69. Polygon(S, T, K) → t4, k_3, s_2, t_2
  70. Point(n_2) → U
  71. Point(k_2) → V
  72. Polygon(U, J, V) → t5, v, u, j_2
  73. Point(l_2) → W
  74. Polygon(S, W, I) → t6, i_3, s_3, w
  75. Angle(L, N, K) → μ

AI Drawing Prompt Reference

Draw a geometric proof of the cosine law, including points A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, and line segments AB, BC, CD, DE, EF, FG, GH, HI, IJ, JK, KL, MN, NO. Start by drawing points A and B, then create points C, D, E, F, G, H, I, J, K, L, M, N, O through rotation and translation, and finally connect these points to form polygons and line segments, demonstrating the geometric proof of the cosine law.
The prompt above was compiled from the original drawing commands and is for reference only.