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Central Symmetry: A Multi-Concept Geometric Demonstration

Central Symmetry: A Multi-Concept Geometric Demonstration
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2020-04-08 17:00
CirclePolygonAngleTransformationGeometric Construction
This GeoGebra example dynamically illustrates central symmetry, where a collection of polygons, circles, and line segments are mirrored about a central point O, visually revealing the congruence between original and transformed figures. The core mathematical concepts include central symmetry, reflection transformation, properties of polygons and circles, while also integrating Pythagorean theorem, definite integrals, conic sections, and algebraic means, showcasing the interconnection between geometry and algebra. Construction highlights: basic shapes are built using PolyLine, Polygon, Circle, and Segment; the Mirror command applies central symmetry; Sequence generates series of points; Intersect and Angle tools mark intersections and angles; If conditions control dynamic visibility. Suitable for middle and high school geometry lessons, math lab activities, or teacher-training sessions on geometric transformations, helping learners grasp invariants and structural relationships under symmetry.

GeoGebra Commands & Parameters

  1. PolyLine(O_1, P_1, A_2, O_1, Q_1, R_1, S_1, T_1, U_1, V_1, W_1, Z_1) → q_1
  2. Point(q_1) → P_3
  3. PolyLine(I, H, N, M, L, K, J, I, G, F, E, D, C, B, A, O) → t
  4. Point(t) → P1
  5. Polygon(W, Z, C_1, D_1, E_1, F_1, A_1, B_1, V) → poly2, w, z, c_1, d_1, e_1, f_1, a_1, b_1, v
  6. Point(v) → G_1
  7. Point(v) → H_1
  8. PolyLine(A_1, F_1, B_1, A_1, E_1, D_1, C_1, Z, W, H_1, G_1, V) → p_1
  9. Point(p_1) → P2
  10. PolyLine(B_2, C_2, L_2, M_2, B_2, D_2, E_2, F_2, G_2, H_2, I_2, J_2, K_2) → r_1
  11. Point(r_1) → P4
  12. Polygon(A, B, K, L, M, N, O, C, D, E, F, G, H, I, J) → poly1, a_2, j, k, l, m, n, o, b, c, d, e, f, g, h, i
  13. Line(Q, R) → p
  14. If(Motif ≟ 1, P1, If[Motif ≟ 2, P2, If[Motif ≟ 3, P_3, If[Motif ≟ 4, P4, P5]]]) → P
  15. Mirror(P, R) → S
  16. Circle(P, a) → q
  17. Line(P, xAxis) → r
  18. Sequence(Segment[Element[Marque, i], Element[Marque, i + 1]], i, 1, Length[Marque]) → Trait
  19. Polygon(H_1, I_1, J_1, G_1) → poly3, h_1, i_1, j_1, g_1
  20. Mirror(H_1, p) → K_1
  21. Mirror(I_1, p) → L_1
  22. Mirror(J_1, p) → M_1
  23. Mirror(G_1, p) → N_1
  24. Polygon(K_1, L_1, M_1, N_1) → poly3', k_1, l_1, m_1, n_1
  25. Segment(P, S) → s
  26. Intersect(s, p) → N_2
  27. Angle(S, N_2, R) → α
  28. Circle(N_2, 1) → s_1
  29. Intersect(s_1, p) → O_2, P_2
  30. Intersect(s_1, s) → Q_2, R_2
  31. Line(R_2, p) → t_1
  32. Line(O_2, s) → b_2
  33. Intersect(b_2, t_1) → S_2
  34. Polygon(N_2, R_2, S_2, O_2) → poly4, n_2, r_2, s_2, o_2
  35. Mirror(Trait, R) → Trait'
  36. Mirror(H_1, R) → T_2
  37. Mirror(I_1, R) → U_2
  38. Mirror(J_1, R) → V_2
  39. Mirror(G_1, R) → W_2
  40. Polygon(T_2, U_2, V_2, W_2) → poly3'_1, t_2, u_2, v_2, w_2
  41. Segment(P, R) → c_2
  42. Segment(S, R) → c_3