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Visual Demonstration of Law of Cosines

Visual Demonstration of Law of Cosines
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2020-01-06 05:30
TriangleTrigonometryGeometric Construction
This example visually demonstrates the Law of Cosines in arbitrary triangles through dynamic geometry. It involves core concepts like the Law of Cosines, triangle side-angle relationships, and vector operations. Construction highlights include using GeoGebra commands such as Line, Polygon, OrthogonalLine, Intersect, Segment, Angle, Ray, Rotate, Circle, and Midpoint to build triangles and auxiliary lines, dynamically adjusting sides and angles to illustrate the theorem's generality. Suitable for high school mathematics instruction and geometric theorem exploration.

GeoGebra Commands & Parameters

  1. Line(B, A) → d
  2. Polygon(B, A, 4) → poly4, r, s, t, a_1, K, L
  3. Line(L, K) → b_1
  4. Line(C, A) → e
  5. OrthogonalLine(B, e) → g
  6. Intersect(e, g) → F
  7. Polygon(A, C, 4) → poly3, m, n, p, q, I, J
  8. Line(I, J) → c_1
  9. Intersect(g, c_1) → N
  10. Polygon(A, F, N, J) → poly6, a_3, f_1, n_1, j_1
  11. Polygon(C, I, N, F) → poly10, c_3, i_1, n_2, f_2
  12. OrthogonalLine(C, d) → f
  13. Intersect(d, f) → E
  14. Intersect(f, b_1) → M
  15. Polygon(A, E, M, K) → poly5, a_2, e_1, m_1, k_1
  16. Polygon(B, E, M, L) → poly9, b_3, e_2, m_2, l_1
  17. Polygon(C, B, 4) → poly2, i, j, k, l, G, H
  18. Line(V, K_1) → t_2
  19. Line(Z, D_1) → e_3
  20. Intersect(t_2, e_3) → L_1
  21. Polygon(A, B, C) → poly1, c, a, b
  22. OrthogonalLine(A, a) → h
  23. Intersect(h, a) → D
  24. Intersect(h, k) → O
  25. Polygon(D, C, H, O) → poly7, d_1, c_2, h_1, o_2
  26. Polygon(D, O, G, B) → poly8, d_2, o_1, g_1, b_2
  27. Segment(A, D) → p_1
  28. Segment(C, E) → q_1
  29. Segment(B, F) → r_1
  30. Angle(B, A, C) → α
  31. Segment(A, E) → s_1
  32. Segment(A, F) → t_1
  33. Ray(B, C) → g_2
  34. Ray(C, B) → h_2
  35. Rotate(C, (-((180° - setalpha) / 2)), C) → C'
  36. Rotate(B, (-((180° - (2 * setalpha)) / 2)), C) → B'
  37. Ray(C', B') → i_2
  38. Rotate(B, (180° - (2 * setalpha)) / 2, B) → B'_1
  39. Rotate(C, (180° - (2 * setalpha)) / 2, B) → C'_1
  40. Ray(B'_1, C'_1) → j_2
  41. Intersect(i_2, j_2) → P
  42. Circle(P, C) → k_2
  43. Ray(D, A) → l_2
  44. Intersect(k_2, l_2) → Q,
  45. Midpoint(B, C) → R
  46. Circle(R, C) → p_2
  47. Intersect(p_2, l_2) → S, C_1
  48. Polygon(T, J_1, I_1, H_1) → poly16, t_3, j_3, i_4, h_4
  49. Polygon(H_1, I_1, K_1, V) → poly11, h_3, i_3, k_3, v
  50. Line(W, F_1) → l_3
  51. Intersect(l_3, v) → M_1
  52. Polygon(H_1, I_1, W, M_1) → poly15, h_5, i_5, w, m_3
  53. Polygon(H_1, V, K_1, I_1) → poly17, n_3, p_3, q_3, r_3
  54. Polygon(H_1, I_1, W, M_1) → poly18, h_6, i_6, w_1, m_4
  55. Polygon(M_1, W, K_1, V) → poly19, m_5, w_2, k_4, v_1
  56. Polygon(W, F_1, L_1, K_1) → poly20, q_2, r_2, s_2, f_3
  57. Polygon(U, Z, B_1, A_1) → poly12, u_1, z, b_4, a_4
  58. Polygon(A_1, B_1, N_1, G_1) → poly13, d_3, s_3, c_4, e_4
  59. Polygon(K_1, G_1, N_1, D_1) → poly14, k_5, g_3, n_4, d_4
  60. Line(G_1, N_1) → f_4
  61. Line(T, V) → g_4
  62. Intersect(f_4, g_4) → O_1
  63. Segment(M_1, O_1) → j_4
  64. Segment(O_1, G_1) → l_4
  65. Midpoint(B, H) → Q_1
  66. Midpoint(J, C) → R_1
  67. Midpoint(K, B) → S_1
  68. Midpoint(A_1, Z) → T_1
  69. Midpoint(t_1) → U_1
  70. Midpoint(F_1, K_1) → V_1
  71. Midpoint(D_1, G_1) → W_1
  72. Midpoint(J, F) → Z_1
  73. Midpoint(K, E) → A_2
  74. Midpoint(T, W) → P_1
  75. Segment(H_1, I_1) → p_4
  76. Polygon(L, K, A, B) → poly21, l_5, k_6, a_5, b_5
  77. Midpoint(B_1, G_1) → C_2
  78. Polygon(J, I, C, A) → poly22, j_5, i_7, c_5, a_6
  79. Midpoint(F_1, D_1) → D_2
  80. Midpoint(D_1, N_1) → E_2

AI Drawing Prompt Reference

Draw an arbitrary triangle to visually demonstrate the Law of Cosines. Include the triangle's three sides and angles, with auxiliary lines highlighting the key relationships in the theorem. Use perpendicular lines, intersection points, and rotation to construct auxiliary shapes, showing the theorem's generality as sides and angles change. The final image should clearly illustrate the connection between the triangle's parts and the Law of Cosines, suitable for high school mathematics instruction and geometric theorem exploration.
The prompt above was compiled from the original drawing commands and is for reference only.