Geometric Intuition for π and a Definite-Integral Setup (V2)

GeoGebra Commands & Parameters

1. Rotate(N, If[1 < a ≤ 2, (-(a - 1)), -1], M) → N'
2. CircleArc(M, N', N) → c
3. Mirror(c, yAxis) → c'
4. Rotate(C, If[1 < a ≤ 2, (-(a - 1)), -1], N') → C'
5. Rotate(C', -2, M) → C''_{1}
6. Rotate(C''_{1}, -2, M) → O
7. Line(O, C''_{1}) → t
8. Line(M, t) → g_1
9. Intersect(c', g_1) → H,
10. Line(C', C''_{1}) → h
11. Line(M, h) → i
12. Intersect(c, i, 1) → D
13. Angle(D, M, H) → β
14. Rotate(D, If[5 < a ≤ 6, ((a - 5) * β), β], M) → D'
15. Mirror(D', M) → D''
16. Segment(D', D'') → s_1
17. Circle(H, s_1 / 7) → t_1
18. Intersect(t_1, s_1, 1) → R
19. Mirror(H, R) → H'
20. Mirror(R, H') → R'
21. Mirror(H', R') → H''
22. Mirror(C', yAxis) → C''
23. Line(M, C'') → n_2
24. OrthogonalLine(C'', n_2) → p_2
25. Line(M, p_2) → q_2
26. Point(q_2) → U
27. Segment(((-r), 0), (r, 0)) → f
28. Rotate(B, If[7.5 < a ≤ 8.5, ((-(a - 7.5)) * 2), -2], (0, 0)) → B'_{2}
29. Line((0, 0), B'_{2}) → b
30. OrthogonalLine(B'_{2}, b) → f_1
31. Circle((0, 0), r) → d_1
32. Vector(I, J) → v
33. Rotate((r, 0), 1, (0, 0)) → K
34. Rotate((r, 0), 3, (0, 0)) → L
35. Mirror(K, y = 0) → K'
36. Dilate(M, If[0 < a ≤ 1, 1 - a, 0], N) → M'
37. Vector(M, M') → w
38. Translate(f, w) → f'_{2}
39. Mirror(N', yAxis) → N''
40. Mirror(A, yAxis) → A'
41. Circle(N', r - c) → d
42. Segment(N', C') → g
43. Mirror(N', yAxis) → N''_{1}
44. Segment(N''_{1}, C'') → g'
45. Rotate(C', -1, M) → C''_{2}
46. Angle(B, M, D) → α
47. Rotate(B, If[2 < a ≤ 3, ((a - 2) * α), α], M) → B'
48. Mirror(B', M) → B''
49. Segment(B'', B') → j
50. Dilate(M, If[3 < a ≤ 4, 4 - a, 0], C''_{2}) → M'_{1}
51. Vector(M, M'_{1}) → u
52. Translate(B'', u) → B'''
53. Translate(B', u) → B''_{1}
54. Segment(B''', B''_{1}) → j'
55. Rotate(C''_{2}, If[4 < a ≤ 5, a - 4, 1], M) → E
56. Rotate(C''_{2}, If[4 < a ≤ 5, (-(a - 4)), -1], M) → F
57. CircleArc(M, C''_{2}, E) → e
58. CircleArc(M, F, C''_{2}) → k
59. Circle(E, j / 2 - e) → p
60. Line(M, E) → l
61. OrthogonalLine(E, l) → m
62. Rotate(M, ((-pi)) / 2, E) → M'_{2}
63. Ray(E, M'_{2}) → n
64. Intersect(p, n, 1) → G
65. Rotate(M, pi / 2, E) → M'_{3}
66. Segment(E, G) → q
67. Line(M, C''_{2}) → s
68. Mirror(E, s) → E'
69. Mirror(G, s) → G'
70. Segment(E', G') → q'
71. Rotate(C''_{1}, -1, M) → P
72. Segment(D', D'') → h_1
73. Dilate(M, If[6 < a ≤ 7, 7 - a, 0], P) → M'_{4}
74. Vector(M, M'_{4}) → u_1
75. Translate(D', u_1) → D''_{1}
76. Translate(D'', u_1) → D'''
77. Segment(D''_{1}, D''') → i_1
78. Rotate(P, If[7 < a ≤ 8, a - 7, 1], M) → P'
79. Line(M, P) → j_1
80. Mirror(P', j_1) → P''
81. CircleArc(M, P, P') → c_1
82. CircleArc(M, P'', P) → e_1
83. Circle(P', h_1 / 2 - c_1) → k_1
84. Circle(P'', h_1 / 2 - c_1) → p_1
85. Line(M, P') → l_1
86. OrthogonalLine(P', l_1) → m_1
87. Rotate(M, ((-pi)) / 2, P') → M'_{5}
88. Ray(P', M'_{5}) → n_1
89. Intersect(k_1, n_1) → Q,
90. Segment(P', Q) → q_1
91. Mirror(Q, j_1) → Q'
92. Segment(P'', Q') → r_1
93. Mirror(R', H'') → R''
94. Mirror(H'', R'') → H'''
95. Mirror(R'', H''') → R'''
96. Dilate(H, If[8 < a ≤ 9, 9 - a, 0], R) → H'_{1}
97. Segment(H, H'_{1}) → a_1
98. Rotate(H, If[9 < a ≤ 10, ((a - 9) * pi), pi], R) → H'_{2}
99. Segment(R, H'_{2}) → b_1
100. Rotate(R, If[10 < a ≤ 11, ((a - 10) * pi), pi], H') → R'_{1}
101. Segment(H', R'_{1}) → f_2
102. Rotate(H', If[11 < a ≤ 12, ((a - 11) * pi), pi], R') → H''_{1}
103. Segment(R', H''_{1}) → g_2
104. Rotate(R', If[12 < a ≤ 13, ((a - 12) * pi), pi], H'') → R''_{1}
105. Segment(H'', R''_{1}) → h_2
106. Rotate(H'', If[13 < a ≤ 14, ((a - 13) * pi), pi], R'') → H'''_{1}
107. Segment(R'', H'''_{1}) → i_2
108. Rotate(R'', If[14 < a ≤ 15, ((a - 14) * pi), pi], H''') → R'''_{1}
109. Segment(H''', R'''_{1}) → j_2
110. Line(M, C'') → k_2
111. Angle(U, M, H'') → γ
112. Dilate(H'', If[15.5 < a ≤ 16.5, 16.5 - a, 0], C'') → H'''_{2}
113. Vector(H'', H'''_{2}) → v_1
114. Dilate(R', If[15.5 < a ≤ 16.5, 16.5 - a, 0], C'') → R''_{2}
115. Vector(R', R''_{2}) → w_1
116. Translate(R', w_1) → R''_{3}
117. Translate(H''_{1}, w_1) → S
118. Segment(R''_{3}, S) → l_2
119. Rotate(R''_{3}, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * γ), (-γ)], R''_{2}) → T
120. Rotate(S, If[15.5 < a ≤ 16.5, ((-(a - 15.5)) * γ), (-γ)], R''_{2}) → S'
121. Segment(T, S') → m_2
122. Rotate(C'', If[16.5 < a ≤ 17.5, ((a - 16.5) * 0.28), 0.28], M) → C'''
123. CircleArc(M, C'', C''') → c_2
124. Circle(C''', (0.14 * s_1) - c_2) → d_2
125. Line(C''', M) → r_2
126. OrthogonalLine(C''', r_2) → s_2
127. Rotate(M, ((-pi)) / 2, C''') → M'_{6}
128. Ray(C''', M'_{6}) → t_2
129. Intersect(d_2, t_2, 1) → V
130. Segment(C''', V) → a_2
131. If(8 < a ≤ 15, 0.75, 0.25) → o_1