Parallelogram and 2 Equilateral Triangles: How to Prove?

GeoGebra Commands & Parameters

1. Rotate(B, a°, A) → B'
2. Angle(B, A, B') → α
3. Ray(A, B) → f
4. Ray(A, B') → g
5. Line(B, g) → h
6. Point(g) → C
7. Line(C, f) → i
8. Intersect(h, i) → D
9. Segment(C, D) → j
10. Segment(A, B) → k
11. Segment(B, D) → l
12. Segment(A, C) → m
13. Min(m, k) → b
14. Circle(C, (0.2 * b)) → d
15. Circle(A, (0.2 * b)) → e
16. Circle(B, (0.2 * b)) → p
17. Circle(D, (0.2 * b)) → c
18. Intersect(e, k, 1) → E
19. Intersect(e, m, 1) → F
20. Intersect(p, k, 1) → G
21. Intersect(p, l, 1) → H
22. Intersect(d, g, 2) → I
23. Ray(C, I) → q
24. Intersect(c, h, 2) → J
25. Ray(D, J) → r
26. Intersect(c, i, 2) → K
27. Ray(D, K) → s
28. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], B) → A'
29. Segment(A, A') → t
30. Rotate(E, If[1 < o ≤ 2, ((o - 1) * α), α], A) → E'
31. CircleSector(A, E, E') → c_1
32. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], C) → A'_1
33. Vector(A, A'_1) → u
34. Translate(c_1, u) → d_1
35. Segment(A, A'_1) → f_1
36. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], D) → C'
37. Vector(C, C') → v
38. Translate(d_1, v) → e_1
39. Segment(C, C') → g_1
40. Dilate(D, If[4 < o ≤ 5, 5 - o, 0], B) → D'
41. Segment(D, D') → h_1
42. Polygon(A, B, D, C) → poly1, a_1, b_1, d_2, c_2
43. Dilate(A, If[5.5 < o ≤ 6.5, 6.5 - o, 0], C) → A'_2
44. Vector(A, A'_2) → w
45. Translate(A, w) → A'_3
46. Translate(A', w) → A''
47. Dilate(A, If[6.5 < o ≤ 7.5, 7.5 - o, 0], B) → A'_4
48. Vector(A, A'_4) → u_1
49. Translate(A, u_1) → A'_5
50. Translate(A'_1, u_1) → L
51. Rotate(B, π / 3, D) → B'_1
52. Rotate(D, π / 3, C) → D'_1
53. Rotate(D, If[6 < o ≤ 7, ((o - 6) * π / 3), π / 3], D) → D'_2
54. Rotate(D', If[6 < o ≤ 7, ((o - 6) * π / 3), π / 3], D) → D''
55. Segment(D'_2, D'') → i_1
56. Rotate(C, If[6 < o ≤ 7, ((o - 6) * π / 3), π / 3], C) → C'_1
57. Rotate(C', If[6 < o ≤ 7, ((o - 6) * π / 3), π / 3], C) → C''
58. Segment(C'_1, C'') → j_1
59. Polygon(D'_1, B'_1, A) → t1, a_2, d'_1, b'_1
60. Rotate(C, If[7 < o ≤ 8, ((-(o - 7)) * π / 3), ((-π)) / 3], C') → C'_2
61. Rotate(C', If[7 < o ≤ 8, ((-(o - 7)) * π / 3), ((-π)) / 3], C') → C''_1
62. Segment(C'_2, C''_1) → k_1
63. Rotate(D, If[7 < o ≤ 8, ((-(o - 7)) * π / 3), ((-π)) / 3], B) → D'_3
64. Rotate(D', If[7 < o ≤ 8, ((-(o - 7)) * π / 3), ((-π)) / 3], B) → D''_1
65. Segment(D'_3, D''_1) → l_1
66. Polygon(C, C', C'') → t2, c'', c_3, c'
67. Polygon(C', D'', B) → t3, b_2, c'_1, d''
68. Dilate(C'', If[8 < o ≤ 9, 9 - o, 0], D'') → C'''
69. Segment(C'', C''') → m_1
70. Rotate(C'', If[9 < o ≤ 10, ((o - 9) * π / 3), π / 3], D'') → C'''_1
71. Rotate(C''', If[9 < o ≤ 10, ((o - 9) * π / 3), π / 3], D'') → M
72. Rotate(C'', If[9 < o ≤ 10, ((o - 9) * π / 3), π / 3], D'') → C'''_2
73. Rotate(C''', If[9 < o ≤ 10, ((o - 9) * π / 3), π / 3], D'') → N
74. Segment(C'''_2, N) → n_1
75. Rotate(C'''_2, If[10 < o ≤ 11, ((o - 10) * π / 3), π / 3], A) → O
76. Rotate(N, If[10 < o ≤ 11, ((o - 10) * π / 3), π / 3], A) → N'
77. Segment(O, N') → p_1