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AA Similarity Theorem Proof with GeoGebra

AA Similarity Theorem Proof with GeoGebra
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2020-12-02 21:14
TriangleAngleTransformationGeometric Construction
This example demonstrates the Angle-Angle Triangle Similarity Theorem ('Proof') using transformational geometry tools. It involves core math concepts like angles, congruence, similar triangles, triangles, polynomial functions, binomial distribution, division, incircle or inscribed circle, and pyramid. Constructed with commands like Segment, Circle, Point, Angle, Rotate, Ray, Intersect, Line, Dilate, Vector, Polygon, Translate, and If. Suitable for high school mathematics teaching or self-learning to understand and apply the AA Similarity Theorem.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. Circle(D, (k * a)) → d
  5. Point(d) → E
  6. Segment(D, E) → e
  7. Circle(E, (k * b)) → f
  8. Angle(A, B, C) → α
  9. Rotate(D, α, E) → D'
  10. Angle(D, E, D') → β
  11. Ray(E, D') → g
  12. Intersect(f, g, 1) → F
  13. Angle(C, A, B) → γ
  14. Rotate(E, (-γ), D) → E'
  15. Angle(E', D, E) → δ
  16. Ray(D, E') → h
  17. Segment(D, E') → i
  18. Segment(E', E) → j
  19. Line(E, D') → l
  20. Line(E, D') → n
  21. Circle(D, (k * c)) → p
  22. Ray(D, E') → m
  23. Intersect(p, m, 1) → G
  24. Segment(E, G) → q
  25. Segment(D, G) → r
  26. Dilate(D, If[0 < s ≤ 1, 1 - s], A) → D'_1
  27. Vector(D, D'_1) → u
  28. Polygon(A, C, B) → poly1, b_1, a_1, c_1
  29. Polygon(D, E, G) → poly2, g_1, d_1, e_1
  30. Translate(D, u) → D'_2
  31. Translate(E, u) → E'_1
  32. Translate(G, u) → G'
  33. Polygon(D'_2, E'_1, G') → poly2', g', d'_2, e'_1
  34. Vector(D, A) → v
  35. Translate(D, v) → D'_3
  36. Translate(E, v) → E'_2
  37. Translate(G, v) → G'_1
  38. Polygon(D'_3, E'_2, G'_1) → poly2'_1, g'_1, d'_3, e'_2
  39. Segment(D'_1, G') → t
  40. Segment(E'_1, D'_1) → h_1
  41. Segment(A, G'_1) → i_1
  42. Segment(G'_1, E'_2) → j_1
  43. Segment(E'_2, A) → k_1
  44. Angle(C, A, E'_2) → ε
  45. If(ε > γ, ε - γ, γ - ε) → ζ
  46. Rotate(D'_3, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → H
  47. Rotate(E'_2, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → I
  48. Rotate(G'_1, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → J
  49. Polygon(H, I, J) → poly3, j_2, h_2, i_2
  50. Segment(A, J) → l_1
  51. Segment(A, I) → m_1
  52. Segment(I, J) → n_1
  53. Dilate(J, If[2 < s ≤ 3, 3 - s, 0], C) → J'
  54. Dilate(I, If[2 < s ≤ 3, 3 - s, 0], B) → I'
  55. Segment(I', J') → p_1
  56. Angle(A, I', J') → η
  57. Angle(A, I, J) → θ
  58. Angle(D'_2, E'_1, G') → ι
  59. Angle(J, A, I) → λ
  60. Angle(G', D'_1, E'_1) → μ
  61. Polygon(A, J', I') → poly4, i', a_2, j'
  62. Angle(A, I, J) → ν
  63. Segment(G'_1, E'_2) → q_1
  64. Segment(G', E'_1) → f_1
  65. Angle(G'_1, A, E'_2) → κ
  66. Angle(A, E'_2, G'_1) → ξ
  67. Rotate(D'_3, ((s - 1) * ζ), A) → K
  68. Rotate(D'_3, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → L
  69. Rotate(E'_2, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → M
  70. Rotate(G'_1, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → N
  71. Polygon(L, M, N) → poly5, n_2, l_2, m_2
  72. Dilate(M, If[2 < s ≤ 3, 3 - s, 0], B) → M'
  73. Dilate(N, If[2 < s ≤ 3, 3 - s, 0], C) → N'
  74. Segment(M', N') → r_1
  75. Polygon(A, M', N') → poly6, n', a_3, m'
  76. Angle(A, M', N') → ο
  77. Angle(A, M, N) → ρ
  78. Segment(A, M) → t_1
  79. Segment(M, N) → b_2
  80. Segment(A, N) → c_2
  81. Angle(B, A, C) → σ
  82. Angle(G, E, D) → τ
  83. Angle(N, M, A) → υ
  84. Angle(N', M', A) → ϕ

AI Drawing Prompt Reference

Draw two triangles, one original and one constructed to be similar via rotation and scaling. Start with the original triangle's three sides. Then, using a point as center, draw circles with radii proportional to the original sides, pick a point on a circle to form one side of the new triangle. Measure two angles of the original triangle, replicate these angles by rotation onto the new triangle, and find the third vertex via intersection of a ray and a circle. Then copy the entire triangle using scaling and vector translation to create multiple translated copies. Finally, rotate these copies around a vertex to align with the original triangle, demonstrating that equal angles imply similarity.
The prompt above was compiled from the original drawing commands and is for reference only.