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Symmetry Artist: Dynamic Mirror Geometry with Tangents and Intersections

Symmetry Artist: Dynamic Mirror Geometry with Tangents and Intersections
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2019-03-11 19:01
Coordinate GeometryTransformationGeometric ConstructionLocusGraphing
① This interactive GeoGebra example demonstrates how a set of points and segments can be mirrored across an axis (e.g., the y-axis), creating a symmetric pattern while also revealing intersections of tangent lines and special points. ② Core mathematical concepts include axial symmetry, reflection transformation, tangent lines, intersection points, special points (midpoints, feet of perpendiculars), and median lines, blending transformation geometry with coordinate geometry. ③ Construction highlights: using GeoGebra commands Line, Mirror, and Segment, the original points (P_1, A_1…H_1) and their mirrored counterparts (P', A'…H') are generated, along with the symmetry axis axis'. Tangent lines and intersection commands create auxiliary lines (a, b, c, h, i, j) and special points (I_1, J_1…Q_1 and their mirrors), allowing dynamic manipulation. ④ Suitable for high school to early college geometry courses, as well as for artistic symmetric pattern design, helping learners grasp the properties of reflection and the relationships between geometric objects.

GeoGebra Commands & Parameters

  1. Line(X, Y) → axis_y
  2. Mirror(P_1, axis_y) → P'
  3. Mirror(A_1, axis_y) → A'
  4. Mirror(B_1, axis_y) → B'
  5. Mirror(C_1, axis_y) → C'
  6. Mirror(D_1, axis_y) → D'
  7. Mirror(E_1, axis_y) → E'
  8. Mirror(F_1, axis_y) → F'
  9. Mirror(G_1, axis_y) → G'
  10. Mirror(H_1, axis_y) → H'
  11. Line(X, Y) → axis'
  12. Mirror(I_1, axis_y) → I'
  13. Mirror(J_1, axis_y) → J'
  14. Mirror(K_1, axis_y) → K'
  15. Mirror(L_1, axis_y) → L'
  16. Mirror(M_1, axis_y) → M'
  17. Mirror(Q_1, axis_y) → Q'
  18. Segment(P_1, A_1) → a
  19. Segment(A_1, B_1) → b
  20. Segment(B_1, C_1) → c
  21. Segment(G_1, H_1) → h
  22. Segment(H_1, I_1) → i
  23. Segment(I_1, J_1) → j
  24. Segment(J_1, K_1) → k
  25. Segment(K_1, L_1) → l
  26. Segment(L_1, M_1) → m
  27. Segment(M_1, Q_1) → n
  28. Segment(P_1, A') → a_1
  29. Segment(A', B') → b_1
  30. Segment(B', C') → c_1
  31. Segment(G', H') → h_1
  32. Segment(H', I') → i_1
  33. Segment(I', J') → j_1
  34. Segment(J', K') → k_1
  35. Segment(K', L') → l_1
  36. Segment(L', M') → m_1
  37. Segment(M', Q') → n_1
  38. Segment(C_1, D_1) → d
  39. Segment(D_1, E_1) → e
  40. Segment(E_1, F_1) → f
  41. Segment(F_1, G_1) → g
  42. Segment(C', D') → d_1
  43. Segment(D', E') → e_1
  44. Segment(E', F') → f_1
  45. Segment(F', G') → g_1