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Similar Triangles Construction and Properties

Similar Triangles Construction and Properties
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2018-01-02 23:51
TrianglePolygonAngleTransformationGeometric Construction
① This GeoGebra example dynamically demonstrates the construction of similar triangles, visually showing that corresponding angles are equal and corresponding sides are proportional. ② Core mathematical concepts include the criteria for triangle similarity, proportional relationships, and the preservation of shape under transformations such as dilation, reflection, and rotation. ③ Construction highlights: from the summary, commands such as Mirror, Ray, Point, Line, Intersect, Polygon, Segment, Dilate, OrthogonalLine, Angle, and Distance are used to create multiple triangles (poly1, poly2, poly3) and auxiliary lines, generating similar figures through scaling and mirroring. ④ Suitable for middle school or early high school students learning about similar triangles, as well as for teaching geometric construction with dynamic software.

GeoGebra Commands & Parameters

  1. Mirror(G, F) → G'
  2. Ray(F, G') → l
  3. Point(l) → J
  4. Line(H, F) → q_2
  5. Line(G, H) → l_2
  6. Line(J, l_2) → p_2
  7. Intersect(q_2, p_2) → C_1
  8. Line(A, B) → b_1
  9. Polygon(C, A, B) → poly1, b, c, a_2
  10. Ray(A, C) → g
  11. Point(g) → D
  12. Line(D, a_2) → h_2
  13. Intersect(h_2, b_1) → Z
  14. Polygon(F, G, H) → poly2, h_1, f_1, g_1
  15. Ray(A, B) → f
  16. Segment(D, A) → h
  17. Segment(E, A) → i
  18. Segment(D, E) → j
  19. Mirror(H, F) → H'
  20. Ray(F, H') → k
  21. Segment(I, F) → m
  22. Segment(F, J) → n
  23. Segment(J, I) → p
  24. Polygon(A, D, E) → poly3, e, a_1, d
  25. Polygon(F, J, I) → poly4, i_1, f_2, j_1
  26. Polygon(K, L, M) → poly5, m_1, k_1, l_1
  27. Dilate(M, o, M) → M'
  28. Dilate(K, o, Scale) → K'
  29. Dilate(L, o, Scale) → L'
  30. Dilate(M, o, Scale) → M'_1
  31. Polygon(K', L', M'_1) → poly5', m'_1, k', l'
  32. Segment(O, P) → q
  33. Segment(O, Q) → r
  34. Segment(P, Q) → s
  35. Dilate(O, 1 / 3, Q) → O'
  36. Line(O', s) → t
  37. Intersect(q, t) → R
  38. Segment(R, O') → n_1
  39. Segment(O, O') → p_1
  40. Segment(O', Q) → q_1
  41. Segment(R, P) → r_1
  42. Segment(R, O) → s_1
  43. Line(A, C) → t_1
  44. Line(F, p) → c_1
  45. Line(G, m) → d_1
  46. Intersect(d_1, c_1) → S
  47. Line(T, U) → e_1
  48. Point(e_1) → V
  49. OrthogonalLine(V, e_1) → g_2
  50. Point(g_2) → W
  51. Segment(W, U) → i_2
  52. Segment(U, V) → j_2
  53. Segment(W, V) → k_2
  54. OrthogonalLine(V, i_2) → m_2
  55. OrthogonalLine(V, j_2) → n_2
  56. Intersect(m_2, i_2) → A_1
  57. Segment(V, A_1) → r_2
  58. Intersect(i_2, r_2) → B_1
  59. Angle(V, U, W) → β
  60. Angle(U, W, V) → γ
  61. Angle(W, A_1, V) → δ
  62. Angle(U, A_1, V) → ε
  63. Angle(W, V, A_1) → ζ
  64. Angle(U, V, A_1) → η
  65. Distance(W, i_2) → distanceWi2
  66. Distance(A_1, W) → distanceA1W
  67. Distance(A_1, U) → distanceA1U
  68. Line(G, F) → s_2