AA Similarity Theorem: Proof using Transformational Geometry

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