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2D Vectors and Circle Construction by Claudio Martínez Gil

2D Vectors and Circle Construction by Claudio Martínez Gil
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2019-10-06 03:53
CircleVectorCoordinate GeometryGeometric Construction
This example demonstrates the construction of geometric figures using 2D vectors and circles, involving core mathematical concepts such as vectors, circles, and coordinate geometry. Key construction steps include using GeoGebra commands and objects like Line, Polygon, Midpoint, OrthogonalLine, Vector, OrthogonalVector, Intersect, Mirror, Circle, CurveCartesian, CircleArc, Semicircle, Segment. Suitable for high school and above students learning vectors, geometric constructions, and coordinate geometry.

GeoGebra Commands & Parameters

  1. Line(A, B) → m
  2. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  3. Midpoint(D, B) → G
  4. OrthogonalLine(G, m) → n
  5. Midpoint(D, A) → E
  6. Line(E, G) → j
  7. Vector(A, B) → u
  8. OrthogonalVector(u) → v
  9. Line(K, j) → r
  10. Midpoint(A, B) → J
  11. OrthogonalLine(H, f) → q
  12. Intersect(q, r) → H_2
  13. Mirror(H, n) → H'
  14. OrthogonalLine(H', m) → b
  15. Intersect(b, r) → G_2
  16. Mirror(J_2, n) → Z_2
  17. Line(L, j) → k
  18. OrthogonalLine(F, m) → l
  19. Intersect(l, j) → I
  20. Mirror(F, n) → F'
  21. OrthogonalLine(H, m) → p
  22. OrthogonalLine(F', m) → a
  23. Mirror(I, n) → I'
  24. Line(C, B) → g_2
  25. Line(M, j) → s
  26. Line(N, j) → t
  27. Midpoint(M, N) → O
  28. Line(O, j) → c
  29. Intersect(n, c) → P
  30. Intersect(c, p) → Q
  31. Line(R, q) → e
  32. Intersect(c, e) → S
  33. Circle(S, P) → d
  34. Circle(S, T) → c_1
  35. Intersect(k, l) → U
  36. Line(V, j) → f_1
  37. Intersect(l, f_1) → W
  38. Intersect(n, h) → Z
  39. CurveCartesian(((1 - t)^(3) * W) + (((3 * (1 - t)^(2)) * t) * A_1) + (((3 * (1 - t)) * t^(2)) * K) + (t^(3) * U), t, 0, 1) → w
  40. CurveCartesian(((1 - t)^(3) * W) + (((3 * (1 - t)^(2)) * t) * B_1) + (((3 * (1 - t)) * t^(2)) * C_1) + (t^(3) * Z), t, 0, 1) → a_1
  41. Intersect(e, k) → D_1
  42. CurveCartesian(((1 - t)^(3) * U) + (((3 * (1 - t)^(2)) * t) * G_1) + (((3 * (1 - t)) * t^(2)) * F_1) + (t^(3) * E_1), t, 0, 1) → b_1
  43. Intersect(s, w) → H_1, I_1, J_1
  44. Intersect(t, w) → K_1, L_1, M_1
  45. Intersect(d, t) → N_1, O_1
  46. Intersect(d, s) → P_1, Q_1
  47. CurveCartesian(((1 - t)^(4) * V_1) + (((4 * (1 - t)^(3)) * t) * Z_1) + (((6 * (1 - t)^(2)) * t^(2)) * A_2) + (((4 * (1 - t)) * t^(3)) * W_1) + (t^(4) * T_1), t, 0, 1) → d_1
  48. CurveCartesian(((1 - t)^(4) * S_1) + (((4 * (1 - t)^(3)) * t) * B_2) + (((6 * (1 - t)^(2)) * t^(2)) * C_2) + (((4 * (1 - t)) * t^(3)) * D_2) + (t^(4) * U_1), t, 0, 1) → e_1
  49. Circle(E_2, F_2) → g_1
  50. CircleArc(G, H_2, G_2) → h_1
  51. CircleArc(G, I_2, J_2) → k_1
  52. Polygon(U_1, E_1, J, R_1) → q1, u_1, i_1, j_1, r_1
  53. Polygon(Z, W, T_1, V_1) → q2, z_1, w_1, t_1, v_1
  54. Polygon(S_1, U, E_1, U_1) → q3, s_1, u_2, l_1, m_1
  55. Polygon(T_1, N_1, P_1, S_1) → q4, n_1, p_1, q_1, f_2
  56. Polygon(I_2, J_2, G_2, H_2) → q5, i_2, j_2, h_2, k_2
  57. Semicircle(H_2, C_3) → c_2
  58. Mirror(c_2, n) → d_2
  59. Mirror(c_1, n) → e_2
  60. Mirror(g_1, n) → p_2
  61. Mirror(d, n) → d'
  62. Mirror(a_1, n) → o_1
  63. Mirror(b_1, n) → a_2
  64. Mirror(w, n) → w'
  65. Mirror(U_1, n) → K_2
  66. Mirror(E_1, n) → L_2
  67. Mirror(J, n) → J'
  68. Mirror(R_1, n) → M_2
  69. Polygon(K_2, L_2, J', M_2) → q1', l_2, m_2, j', n_2
  70. Mirror(Z, n) → Z'
  71. Mirror(W, n) → W'
  72. Mirror(T_1, n) → N_2
  73. Mirror(V_1, n) → O_2
  74. Polygon(Z', W', N_2, O_2) → q2', z', q_2, r_2, o_2
  75. Mirror(S_1, n) → P_2
  76. Mirror(U, n) → U'
  77. Mirror(E_1, n) → Q_2
  78. Mirror(U_1, n) → R_2
  79. Polygon(P_2, U', Q_2, R_2) → q3', s_2, u', t_2, b_2
  80. Mirror(T_1, n) → S_2
  81. Mirror(N_1, n) → T_2
  82. Mirror(P_1, n) → U_2
  83. Mirror(S_1, n) → V_2
  84. Polygon(S_2, T_2, U_2, V_2) → q4', f_3, g_3, h_3, v_2
  85. Mirror(I_2, n) → W_2
  86. Mirror(G_2, n) → A_3
  87. Mirror(H_2, n) → B_3
  88. Polygon(W_2, Z_2, A_3, B_3) → q5', w_2, z_2, a_3, b_3
  89. Mirror(d_1, n) → c_3
  90. Mirror(e_1, n) → d_3
  91. Segment(I_2, W_2) → i_3
  92. Segment(T_1, V_1) → j_3
  93. Segment(S_1, U_1) → k_3
  94. Segment(T_1, O_2) → l_3

AI Drawing Prompt Reference

Draw a geometric figure constructed using 2D vectors and circles, showcasing core mathematical concepts. Start by drawing a line segment and its circumcircle, then find midpoints and perpendicular lines. Use vector operations to create auxiliary lines and circles. Through intersection, reflection, and mirroring, generate key points and curves. Finally, depict the complex geometric pattern using circles and Bézier curves, highlighting the application of vectors and coordinate geometry.
The prompt above was compiled from the original drawing commands and is for reference only.