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Build a Square Workshop: Geometric Construction with Mirror and Angle Bisectors

Build a Square Workshop: Geometric Construction with Mirror and Angle Bisectors
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2017-11-04 08:02
PolygonAngleCoordinate GeometryTransformationGeometric Construction
In collections Elementary Geometry
This GeoGebra example demonstrates how to construct a square and its associated figures step by step, using mirror reflections and angle bisectors to generate rich symmetric structures. The core mathematical concepts include properties of squares, axial symmetry, angle bisectors, similarity transformations, and coordinate geometry involving points and polygons. Based on the construction summary, the key steps are: using the Point command to create essential points A_1 through D_1, the Polygon command to form the square poly4, the Mirror command to reflect edges or diagonals for symmetric copies, and the LineBisector command to construct angle bisectors. These bisectors generate new points (e.g., E_1 to J_1) and polygons (e.g., poly5), leading to a complex network of sub-objects (a_1 to r_1). This example is suitable for middle and high school students learning geometric construction, transformation, and coordinate geometry, and serves as an excellent classroom demonstration for visualizing the interplay between squares, equilateral triangles, and similarity transformations.

GeoGebra Commands & Parameters

  1. Point(Circle[A_1, 3]) → B_1
  2. Polygon(A_1, B_1, C_1, D_1) → q2, a_1, b_1, c_1, d_1
  3. Point(Circle[E_1, 3]) → F_1
  4. Polygon(E_1, F_1, G_1, H_1, I_1, J_1) → poly4, e_1, f_1, g_1, h_1, i_1, j_1
  5. Point(Circle[K_1, 4]) → L_1
  6. Polygon(K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1) → poly5, k_1, l_1, m_1, n_1, o_1, p_1, q_1, r_1
  7. Point(Circle[S_1, 5]) → T_1
  8. Polygon(S_1, T_1, U_1, V_1, W_1, Z_1, A_2, B_2) → poly6, s_1, t_1, u_1, v_1, w_1, z_2, a_2, b_2
  9. Point(Circle[C_2, 3]) → D_2
  10. Polygon(C_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2) → poly7, c_2, d_2, e_2, f_2, g_2, h_2, i_2, j_2
  11. Point(Circle[A, 1]) → B
  12. Polygon(A, B, C, D) → q1, a, b, c, d
  13. Point(Circle[E, 1]) → F
  14. Polygon(E, F, G, H, I, J) → poly1, e, f, g, h, i, j
  15. Point(Circle[K, 2]) → L
  16. Polygon(K, L, M, N, O, P) → poly2, k, l, m, n, o, p
  17. Point(Circle[Q, 3]) → R
  18. Polygon(Q, R, S, T, U, V, W, Z) → poly3, q, r, s, t, u, v, w, z_1
  19. Mirror(E, j) → E'
  20. Mirror(G, j) → G'
  21. Mirror(H, j) → H'
  22. Mirror(I, j) → I'
  23. Mirror(J, j) → J'
  24. Mirror(K, n) → K'
  25. Mirror(L, n) → L'
  26. Mirror(M, n) → M'
  27. Mirror(N, n) → N'
  28. Mirror(O, n) → O'
  29. Mirror(P, n) → P'
  30. Polygon(K', L', M', N', O', P') → poly2', k', l', m', n', o', p'
  31. LineBisector(E_1, F_1) → k_2
  32. Mirror(E_1, k_2) → K_2
  33. Mirror(F_1, k_2) → L_2
  34. Mirror(G_1, k_2) → M_2
  35. Mirror(H_1, k_2) → N_2
  36. Mirror(I_1, k_2) → O_2
  37. Mirror(J_1, k_2) → P_2
  38. Polygon(K_2, L_2, M_2, N_2, O_2, P_2) → poly4', k_3, l_2, m_2, n_2, o_2, p_2
  39. Mirror(K_1, n_1) → Q_2
  40. Mirror(L_1, n_1) → R_2
  41. Mirror(M_1, n_1) → S_2
  42. Mirror(N_1, n_1) → T_2
  43. Mirror(O_1, n_1) → U_2
  44. Mirror(P_1, n_1) → V_2
  45. Mirror(Q_1, n_1) → W_2
  46. Mirror(R_1, n_1) → Z_2
  47. Polygon(Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2) → poly5', q_2, r_2, s_2, t_2, u_2, v_2, w_2, z_3
  48. LineBisector(S_1, T_1) → f_3
  49. Mirror(S_1, f_3) → A_3
  50. Mirror(T_1, f_3) → B_3
  51. Mirror(U_1, f_3) → C_3
  52. Mirror(V_1, f_3) → D_3
  53. Mirror(W_1, f_3) → E_3
  54. Mirror(Z_1, f_3) → F_3
  55. Mirror(A_2, f_3) → G_3
  56. Mirror(B_2, f_3) → H_3
  57. Polygon(A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3) → poly6', a_3, b_3, c_3, d_3, e_3, f_4, g_3, h_3
  58. LineBisector(C_2, D_2) → i_3
  59. Mirror(C_2, i_3) → I_3
  60. Mirror(D_2, i_3) → J_3
  61. Mirror(E_2, i_3) → K_3
  62. Mirror(F_2, i_3) → L_3
  63. Mirror(G_2, i_3) → M_3
  64. Mirror(H_2, i_3) → N_3
  65. Mirror(I_2, i_3) → O_3
  66. Mirror(J_2, i_3) → P_3
  67. Polygon(I_3, J_3, K_3, L_3, M_3, N_3, O_3, P_3) → poly7', i_4, j_3, k_4, l_3, m_3, n_3, o_3, p_3