Proof of Double Angle Trig Identity via Equal Areas

GeoGebra Commands & Parameters

1. Segment((0, 0), (0, 1)) → i
2. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], B) → A'
3. Segment(A, A') → j
4. Dilate(A, If[3 < b ≤ 4, 4 - b, 0], E) → A'_{1}
5. Vector(A, A'_{1}) → u
6. Line(A, E) → k
7. Translate(A, u) → A'_{2}
8. Translate(A', u) → A''
9. Segment(A'_{2}, A'') → j'
10. Rotate(A'_{2}, If[3 < b ≤ 4, ((b - 3) * pi / 2), pi / 2], A'_{2}) → F
11. Rotate(A'', If[3 < b ≤ 4, ((b - 3) * pi / 2), pi / 2], A'_{2}) → A'''
12. Segment(F, A''') → j''
13. Dilate(A''', If[4 < b ≤ 5, 5 - b, 0], G) → H
14. Vector(A''', H) → v
15. Translate(F, v) → F'
16. Translate(A''', v) → I
17. Segment(F', I) → j'''
18. Dilate(E, If[5 < b ≤ 6, 6 - b, 0], A''') → E'
19. Vector(E, E') → w
20. Translate(i, w) → i'
21. Segment((0, 0), A) → l
22. Dilate(A, If[8 < b ≤ 9, 9 - b, 0], C) → A'_{3}
23. Segment(A, A'_{3}) → m
24. Rotate(A, If[10 < b ≤ 11, ((-(b - 10)) * pi / 2), ((-pi)) / 2], A) → A'_{4}
25. Rotate(A'_{3}, If[10 < b ≤ 11, ((-(b - 10)) * pi / 2), ((-pi)) / 2], A) → J
26. Segment(A'_{4}, J) → m'
27. Rotate(A'_{4}, If[11 < b ≤ 12, ((-(b - 11)) * pi), (-pi)], J) → K
28. Rotate(J, If[11 < b ≤ 12, ((-(b - 11)) * pi), (-pi)], J) → J'
29. Segment(K, J') → m''
30. Dilate(A, If[14 < b ≤ 15, 15 - b, 0], D) → A'_{5}
31. Segment(A, A'_{5}) → n
32. Rotate(A, If[16 < b ≤ 17, ((b - 16) * pi), pi], A) → A'_{6}
33. Rotate(A'_{5}, If[16 < b ≤ 17, ((b - 16) * pi), pi], A) → L
34. Segment(A'_{6}, L) → n'
35. Dilate(A, If[17 < b ≤ 18, 18 - b, 0], K) → A'_{7}
36. Vector(A, A'_{7}) → c
37. Translate(A'_{6}, c) → M
38. Translate(L, c) → L'
39. Segment(M, L') → n''
40. Polygon(G, N, E, E') → q1, g_1, n_1, e, e'
41. Polygon(A, K, L', L) → q2, a_1, k_1, l', l_1
42. Dilate(A, If[18 < b ≤ 19, 19 - b, 0], L) → A'_{8}
43. Vector(A, A'_{8}) → d
44. Translate(A'_{4}, d) → O
45. Translate(J, d) → J'_{1}
46. Segment(O, J'_{1}) → m''_{1}
47. Translate(K, d) → K'
48. Translate(J', d) → J''
49. Segment(K', J'') → m'''
50. Circle(N, m' + m'') → s
51. Ray(E, Mirror[E', yAxis]) → f_1
52. Intersect(s, f_1) → P,
53. Dilate(E, If[20 < b ≤ 21, 21 - b, 0], P) → E'_{1}
54. Vector(E, E'_{1}) → t
55. Translate(E, t) → E'_{2}
56. Translate(E', t) → E''
57. Polygon(G, E'', E'_{1}, N) → q3, g_2, e'', e'_{1}, n_2
58. Line(G, E'') → r
59. Line(N, P) → h_1
60. OrthogonalLine(G, h_1) → i_1
61. Intersect(h_1, i_1) → Q
62. Dilate(N, If[22 < b ≤ 23, 23 - b, 0], Q) → N'
63. Vector(N, N') → u_1
64. Translate(E'_{2}, u_1) → R
65. Polygon(E'', G, N', R) → q4, j_1, g_3, m_1, r_1
66. Dilate(Q, If[24 < b ≤ 25, 25 - b, 0], L) → Q'
67. Vector(Q, Q') → v_1
68. Translate(E'', v_1) → E'''
69. Translate(G, v_1) → G'
70. Translate(N', v_1) → N''
71. Translate(R, v_1) → R'
72. Polygon(E''', G', N'', R') → q4', e''', g', p_1, r'
73. Line(L, r) → q_1
74. Point(q_1) → S
75. Angle(L', L, S) → α
76. Rotate(E''', If[24 < b ≤ 25, ((-(b - 24)) * α), (-α)], N'') → T
77. Rotate(G', If[24 < b ≤ 25, ((-(b - 24)) * α), (-α)], N'') → G''
78. Rotate(N'', If[24 < b ≤ 25, ((-(b - 24)) * α), (-α)], N'') → N'''
79. Rotate(R', If[24 < b ≤ 25, ((-(b - 24)) * α), (-α)], N'') → R''
80. Polygon(T, G'', N''', R'') → q4'', t_1, g'', n''', r''
81. Intersect(f, q, (0.5235987757682062, 0.866025403954346)) → U
82. RotateText(text1, pi / 3 - 0.1) → text2
83. RotateText(text3, pi / 4) → text4
84. RotateText(text5, ((-pi)) / 10) → text6
85. RotateText(text8, pi / 2) → text9
86. Segment((0, 1), (pi / 4, 1)) → f_4