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Dynamic Proof of SAS Triangle Congruence Theorem by Tim Brzezinski

Dynamic Proof of SAS Triangle Congruence Theorem by Tim Brzezinski
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2016-08-01 13:03
TriangleGeometric Construction
In collections Triangle Congruence Unit 4 Triangle Congruence Theorems Triangle Congruence Triangle Congruence
This example demonstrates the dynamic proof of the SAS Triangle Congruence Theorem. It involves core mathematical concepts such as triangle congruence criteria, rotation, reflection, etc. The construction utilizes GeoGebra commands and objects like Circle, Point, Segment, Rotate, Angle, Ray, Intersect, Polygon, CircleSector, Mirror, Line, Dilate, Vector, and Translate. Suitable for high school mathematics teaching and geometric proof learning.

GeoGebra Commands & Parameters

  1. Circle(A, a) → c
  2. Point(c) → B
  3. Segment(A, B) → f
  4. Circle(A, b) → d
  5. Rotate(B, α, A) → B'
  6. Angle(B, A, B') → β
  7. Ray(A, B') → g
  8. Intersect(d, g, 1) → C
  9. Segment(B, C) → h
  10. Segment(A, C) → i
  11. Min(f, i) → e
  12. Circle(A, (0.25 * e)) → k
  13. Intersect(k, f) → D,
  14. Intersect(k, i) → E,
  15. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  16. CircleSector(A, D, E) → p
  17. Mirror(A, h) → A'
  18. Mirror(B, h) → B'_1
  19. Mirror(C, h) → C'
  20. Polygon(A', B'_1, C') → poly1', c', a', b'_1
  21. Mirror(p, h) → p'
  22. Mirror(A, h) → A'_1
  23. Mirror(B, h) → B'_2
  24. Segment(A'_1, B'_2) → f'
  25. Mirror(A, h) → A'_2
  26. Mirror(C, h) → C'_1
  27. Segment(A'_2, C'_1) → i'
  28. Line(C, B) → j
  29. Point(j) → G
  30. Point(j) → H
  31. Ray(B, H) → l
  32. Ray(C, G) → m
  33. Circle(F, a) → q
  34. Point(q) → I
  35. Segment(F, I) → n
  36. Rotate(I, α, F) → I'
  37. Angle(I, F, I') → γ
  38. Circle(F, b) → r
  39. Ray(F, I') → s
  40. Intersect(r, s, 1) → J
  41. Segment(I, J) → t
  42. Segment(F, J) → f_1
  43. Dilate(B, If[0 < u ≤ 1, 1 - u, 0], I) → B'_3
  44. Vector(B, B'_3) → v
  45. Translate(A, v) → A'_3
  46. Translate(B, v) → B'_4
  47. Segment(A'_3, B'_4) → f'_1
  48. Line(B, n) → g_1
  49. Point(g_1) → K
  50. Angle(K, B, A) → δ
  51. Rotate(A'_3, If[1 < u ≤ 2, ((-(u - 1)) * δ), (-δ)], I) → L
  52. Rotate(B'_4, If[1 < u ≤ 2, ((-(u - 1)) * δ), (-δ)], I) → M
  53. Segment(L, M) → h_1
  54. Dilate(A, If[2 < u ≤ 3, 3 - u, 0], F) → A'_4
  55. Vector(A, A'_4) → w
  56. Translate(A, w) → A'_5
  57. Translate(C, w) → C'_2
  58. Segment(A'_5, C'_2) → i'_1
  59. Line(A, f_1) → i_1
  60. Point(i_1) → N
  61. Angle(N, A, C) → ε
  62. Rotate(A'_5, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → O
  63. Rotate(C'_2, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → P
  64. Segment(O, P) → j_1
  65. Translate(p, w) → p'_1
  66. Rotate(p'_1, If[3 < u ≤ 4, ((-(u - 3)) * ε), (-ε)], F) → d_1
  67. Dilate(B, If[4 < u ≤ 5, 5 - u, 0], I) → B'_5
  68. Vector(B, B'_5) → u_1
  69. Translate(A, u_1) → A'_6
  70. Translate(B, u_1) → B'_6
  71. Translate(C, u_1) → C'_3
  72. Polygon(A'_6, B'_6, C'_3) → poly1'_1, c'_3, a'_6, b'_6
  73. Line(B, t) → k_1
  74. Point(k_1) → Q
  75. Angle(Q, B, C) → ζ
  76. Rotate(A'_6, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → R
  77. Rotate(B'_6, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → S
  78. Rotate(C'_3, If[5 < u ≤ 6, ((-(u - 5)) * ε), (-ε)], I) → T
  79. Polygon(R, S, T) → poly2, t_1, r_1, s_1
  80. Segment(B, A) → l_1
  81. Segment(A, C) → m_1
  82. Segment(C, B) → n_1
  83. CircleSector(A, D, E) → e_1