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Lines of Symmetry Investigation: Mirror Symmetry in Polygons and Circles

Lines of Symmetry Investigation: Mirror Symmetry in Polygons and Circles
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2017-11-29 14:30
CirclePolygonAngleTransformationGeometric Construction
This GeoGebra example dynamically explores the lines of symmetry of polygons and associated circles (incircle, circumcircle). It demonstrates axial symmetry, reflection, angle bisectors, perpendicular bisectors, and how symmetry axes relate to vertices, edges, and centers. Construction uses commands: Polygon, OrthogonalLine, LineBisector, AngularBisector, Mirror, Intersect, Circle, Segment, Length, Midpoint, Point. Users can drag vertices or adjust sliders to observe changes in the number and position of symmetry lines. Suitable for middle and high school students learning geometric transformations and symmetry, and for teachers to illustrate concepts in class.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 3) → poly1, f, g, h, C
  2. OrthogonalLine(C, f) → i
  3. OrthogonalLine(A, g) → j
  4. Intersect(i, j) → D
  5. Line(E, F) → k
  6. Circle(D, Segment[A, B]) → c
  7. Intersect(c, k, 1) → G
  8. Intersect(c, k, 2) → H
  9. Segment(H, G) → l
  10. Mirror(poly1, l) → poly1'
  11. Polygon(I, J, K) → t1, k_1, i_1, j_1
  12. AngularBisector(J, I, K) → m
  13. AngularBisector(I, K, J) → n
  14. Intersect(m, n) → L
  15. Length(i_1) → a
  16. Length(j_1) → b
  17. Length(k_1) → d
  18. Circle(L, Max[a, Max[b, d]]) → e
  19. Line(M, N) → p
  20. Intersect(e, p, 1) → O
  21. Intersect(e, p, 2) → P
  22. Segment(O, P) → q
  23. Mirror(I, q) → I'
  24. Mirror(J, q) → J'
  25. Mirror(K, q) → K'
  26. Polygon(I', J', K') → t1', k', i', j'
  27. Polygon(Q, R, 4) → poly2, s, t, f_1, g_1, S, T
  28. Midpoint(Q, T) → U
  29. Midpoint(T, S) → V
  30. OrthogonalLine(U, g_1) → h_1
  31. OrthogonalLine(V, f_1) → l_1
  32. Point(l_1) → W
  33. Circle(W, Segment[R, Q]) → c_1
  34. Line(Z, A_1) → m_1
  35. Intersect(c_1, m_1, 1) → B_1
  36. Intersect(c_1, m_1, 2) → C_1
  37. Segment(B_1, C_1) → n_1
  38. Mirror(poly2, n_1) → poly2'
  39. Segment(D_1, E_1) → p_1
  40. OrthogonalLine(E_1, p_1) → q_1
  41. Circle(E_1, Segment[E_1, D_1]) → d_1
  42. Intersect(d_1, q_1, 2) → F_1
  43. Circle(F_1, Segment[E_1, D_1]) → e_1
  44. Intersect(e_1, q_1, 2) → G_1
  45. OrthogonalLine(G_1, q_1) → r_1
  46. OrthogonalLine(D_1, p_1) → s_1
  47. Intersect(r_1, s_1) → H_1
  48. Polygon(E_1, G_1, H_1, D_1) → q1, e_2, g_2, h_2, d_2
  49. LineBisector(D_1, E_1) → t_1
  50. LineBisector(e_2) → a_1
  51. Point(t_1) → I_1
  52. Circle(I_1, e_2) → c_2
  53. Line(J_1, K_1) → b_1
  54. Intersect(c_2, b_1, 1) → L_1
  55. Intersect(c_2, b_1, 2) → M_1
  56. Segment(L_1, M_1) → f_2
  57. Segment(L_1, M_1) → i_2
  58. Mirror(E_1, i_2) → N_1
  59. Mirror(G_1, i_2) → O_1
  60. Mirror(H_1, i_2) → P_1
  61. Mirror(D_1, i_2) → Q_1
  62. Polygon(N_1, O_1, P_1, Q_1) → q1', n_2, o_1, p_2, q_2
  63. Line(R_1, e_2) → j_2
  64. OrthogonalLine(R_1, j_2) → k_2
  65. Circle(R_1, Segment[E_1, D_1]) → r_2
  66. Intersect(r_2, k_2, 2) → T_1
  67. Line(T_1, j_2) → l_2
  68. Line(R_1, S_1) → m_2
  69. Point(j_2) → U_1
  70. Line(U_1, m_2) → s_2
  71. Intersect(l_2, s_2) → V_1
  72. Intersect(l_2, m_2) → W_1
  73. Polygon(V_1, W_1, R_1, U_1) → q2, v_1, w_1, r_3, u_1
  74. Midpoint(W_1, U_1) → Z_1
  75. Circle(Z_1, Segment[R_1, V_1]) → t_2
  76. Line(A_2, B_2) → a_2
  77. Intersect(t_2, a_2, 2) → C_2
  78. Intersect(t_2, a_2, 1) → D_2
  79. Segment(C_2, D_2) → b_2
  80. Mirror(V_1, b_2) → E_2
  81. Mirror(W_1, b_2) → F_2
  82. Mirror(R_1, b_2) → G_2
  83. Mirror(U_1, b_2) → H_2
  84. Polygon(E_2, F_2, G_2, H_2) → q2', e_3, f_3, g_3, h_3