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Cosine Law Geometric Construction and Verification

Cosine Law Geometric Construction and Verification
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2018-09-23 09:22
TriangleCircleGeometric Construction
This example demonstrates the application of the cosine theorem in triangle geometry using GeoGebra. Key mathematical concepts include triangle side-angle relationships, the cosine theorem formula, and geometric construction techniques. Construction involves using commands like Circle, Intersect, and Polygon to create triangles and verify the theorem. Suitable for middle school students learning geometric proofs and high school students understanding trigonometric applications.

GeoGebra Commands & Parameters

  1. Circle(A_1, 5) → c_4
  2. Intersect(c_4, b_1, 1) → A
  3. Intersect(c_4, b_1, 2) → B
  4. Intersect(c_4, d, 1) → C
  5. Polygon(A, C, B) → poly1, b_2, a_1, c_1
  6. Angle(A, C, B) → α
  7. OrthogonalLine(B, b_2) → e
  8. OrthogonalLine(C, c_1) → f
  9. OrthogonalLine(A, a_1) → g_3
  10. Intersect(e, b_2) → D
  11. Intersect(g_3, a_1) → E
  12. Intersect(f, c_1) → F
  13. Circle(F, c_1) → h_3
  14. Intersect(h_3, f, 2) → G
  15. Segment(F, G) → i
  16. Circle(D, b_2) → k
  17. Intersect(k, e, 2) → H
  18. Segment(D, H) → j
  19. Circle(E, a_1) → p
  20. Intersect(p, g_3, 2) → I
  21. Segment(E, I) → l
  22. Line(H, b_2) → m
  23. Line(G, c_1) → n
  24. Line(I, a_1) → q
  25. Line(C, j) → r
  26. Line(A, j) → s
  27. Line(A, i) → t
  28. Line(B, i) → d_1
  29. Line(B, l) → e_1
  30. Line(C, l) → f_1
  31. Intersect(m, r) → J
  32. Intersect(m, s) → K
  33. Intersect(q, f_1) → L
  34. Intersect(q, e_1) → M
  35. Intersect(n, t) → N
  36. Intersect(n, d_1) → O
  37. Polygon(D, H, J, C) → poly2, d_2, h_1, j_1, c_2
  38. Polygon(C, L, I, E) → poly3, c_3, l_1, i_1, e_2
  39. PointIn(poly2) → Pointpoly2
  40. PointIn(poly3) → Pointpoly3_1
  41. Polygon(A, K, H, D) → poly4, a_2, k_1, h_2, d_3
  42. Polygon(A, F, G, N) → poly5, a_3, f_2, g_1, n_1
  43. PointIn(poly5) → Pointpoly5
  44. PointIn(poly4) → Pointpoly4
  45. Polygon(F, B, O, G) → poly6, f_3, b_3, o, g_2
  46. Polygon(B, M, I, E) → poly7, a_5, m_1, i_2, e_3
  47. PointIn(poly7) → Pointpoly7
  48. PointIn(poly6) → Pointpoly6
  49. PointIn(poly3) → Pointpoly3
  50. Segment(B, D) → p_1
  51. Segment(A, E) → q_1
  52. Segment(F, C) → r_1
  53. Midpoint(h_1) → P
  54. Midpoint(k_1) → Q
  55. Midpoint(g_1) → R
  56. Midpoint(o) → S
  57. Midpoint(l_1) → T
  58. Midpoint(m_1) → U
  59. Segment(P, V) → s_1
  60. Segment(Q, W) → t_1
  61. Segment(R, Z) → j_2
  62. Segment(S, B_1) → k_2
  63. Segment(U, C_1) → l_2
  64. Segment(T, D_1) → m_2
  65. Segment(C, J) → b
  66. Segment(B, M) → a
  67. Segment(A, N) → c