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Construction of Coefficients of i – Geometric Demonstration with Semicircle and

Construction of Coefficients of i – Geometric Demonstration with Semicircle and
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2015-08-02 11:08
CircleTrigonometryComplex NumberTransformationGeometric Construction
①This example demonstrates how the coefficients of the imaginary unit i in complex numbers can be constructed geometrically using a semicircle, segments, polygons, and mirror transformations. ②Core mathematical concepts include complex number representation, trigonometric functions (sine/cosine), ratios, variance, and the interplay between algebraic coefficients and geometric construction. ③Construction details: Based on the summary, the GeoGebra commands used are Semicircle, Segment, Circle, Polygon, Vector, Angle, Point, Distance, Intersect, and Mirror. Key objects include A, B, C, D, c, d, a, b, f, E, F, r, g, G, poly1, b_1, a_1, c_1, g_1, m, H, I, pic1, u_3, v, e, J, h, K, i, α_u, β_v, w, L, j, δ_w, Δ_v, k, l, n, which are arranged to reveal the coefficient structure step by step. ④This is suitable for high school or early college students learning the geometric meaning of complex numbers and trigonometric functions, and can serve as a classroom demonstration or self-guided exploration connecting algebra and geometry.

GeoGebra Commands & Parameters

  1. Semicircle(C, D) → c
  2. Semicircle(D, C) → d
  3. Segment(A, D) → a
  4. Circle(A, 1) → f
  5. Semicircle(E, F) → r
  6. Semicircle(F, E) → g
  7. Polygon(B, A, C, G) → poly1, b_1, a_1, c_1, g_1
  8. Vector((1.414213562373095, 1.414213562373095)) → u_3
  9. Vector((1.79, 0.89)) → v
  10. Segment(G, B) → e
  11. Segment(G, J) → h
  12. Segment(G, K) → i
  13. Angle(B, G, K) → α_u
  14. Angle(B, G, J) → β_v
  15. Vector((0.89, 1.79)) → w
  16. Segment(G, L) → j
  17. Angle(B, G, L) → δ_w
  18. Angle(J, G, L) → Δ_v
  19. Circle(G, K) → k
  20. Point(r) → N
  21. Point(g) → M
  22. Segment(N, A) → p
  23. Segment(A, M) → q
  24. Segment(M, N) → s
  25. Polygon(N, C, G, B, M) → poly2, n_1, c_2, g_2, b_2, m_1
  26. Vector((2, 1)) → t
  27. Vector((1, 2)) → u
  28. Segment(G, N) → d_1
  29. Segment(G, M) → e_1
  30. Angle(M, G, N) → α_t
  31. Segment(A, R) → k_1
  32. Segment(A, Q) → l_1
  33. Segment(Q, R) → p_1
  34. Circle(A, S) → q_1
  35. Distance(G, A) → distanceGA
  36. Point(Circle[A, 3]) → T
  37. Segment(A, T) → r_1
  38. Point(Circle[G, 3]) → U
  39. Segment(G, U) → s_1
  40. Point(Circle[T, 2.8284271247461903]) → V
  41. Segment(T, V) → t_1
  42. Distance(G, Q) → distanceGQ
  43. Distance(G, M) → distanceGM
  44. Segment(G, M) → a_2
  45. Segment(W, Z) → d_2
  46. Vector(A_1, B_1) → u_2
  47. Vector(C_1, D_1) → v_2
  48. Distance(G, N) → distanceGN
  49. Segment(G, N) → f_2
  50. Distance(G, Q) → j_2
  51. Point(k) → E_1
  52. Point(k) → F_1
  53. Distance(F_1, E_1) → distanceF1E1
  54. Segment(E_1, F_1) → k_2
  55. Point(k_2) → G_1
  56. Point(s) → H_1
  57. Distance(G_1, H_1) → distanceG1H1
  58. Segment(G_1, H_1) → l_2
  59. Segment(I_1, J_1) → m_2
  60. Segment(K_1, L_1) → n_2
  61. Segment(K_2, L_2) → n_3
  62. Segment(K_3, L_3) → n_4
  63. Segment(K_4, L_4) → n_5
  64. Segment(K_5, L_5) → n_6
  65. Segment(K_8, L_8) → n_9
  66. Polygon(M_1, N_1, O_1, P_1) → poly3, m_3, n_{13}, o_2, p_2
  67. Intersect(k, q_1, 1) → Q_1
  68. Point(f_2) → R_1
  69. Circle(Q_1, R_1) → q_2
  70. Mirror(q_2, i) → r_2
  71. Segment(S_1, T_1) → a_3
  72. Segment(U_1, V_1) → b_3