Area of a Trapezoid (3): Dynamic Construction and Formula Derivation

GeoGebra Commands & Parameters

1. Point(Ray[A, A + (1, 0)]) → B
2. PointIn(d) → C
3. Polygon(A, B, C) → t1, f, a, b
4. OrthogonalLine(C, g) → h
5. Intersect(g, h) → C_{1}
6. Point(Ray[C, C + (1, 0)]) → D
7. Dilate(C, If[0 < c ≤ 1, 1 - c, 0], D) → F'
8. Dilate(D, If[1 < c ≤ 2, 2 - c, 0], C_{1}) → D'
9. Rotate(C, If[2 < c ≤ 3, ((c - 2) * π), π], D) → F'_{1}
10. Rotate(F', If[2 < c ≤ 3, ((c - 2) * π), π], D) → F''
11. Angle(C, B, A) → β
12. Angle(B, A, C) → γ
13. If(γ ≟ π / 2, 12, 16) → o
14. Midpoint(B, D) → E
15. Polygon(A, B, E, C) → q1, a_1, b_1, e, c_1
16. Polygon(E, D, C) → t2, c_2, e_1, d_1
17. Dilate(D, If[0 < c ≤ 1, 1 - c, 0], E) → D'_{1}
18. Segment(D, D'_{1}) → i
19. Rotate(D, If[1 < c ≤ 2, ((-(c - 1)) * π), (-π)], E) → D'_{2}
20. Rotate(D'_{1}, If[1 < c ≤ 2, ((c - 1) * π), π], E) → F
21. Segment(D'_{2}, F) → i'
22. Segment(C, D) → j
23. Segment(A, C) → k
24. Dilate(C, If[2 < c ≤ 3, 3 - c, 0], E) → C'
25. Segment(C, C') → l
26. Rotate(E, If[4 < c ≤ 5, ((-(c - 4)) * π), (-π)], E) → E'
27. Rotate(D, If[4 < c ≤ 5, ((-(c - 4)) * π), (-π)], E) → D'_{3}
28. Rotate(C, If[4 < c ≤ 5, ((-(c - 4)) * π), (-π)], E) → C'_{1}
29. Polygon(E', D'_{3}, C'_{1}) → t2', c'_{1}, e', d'_{3}
30. Segment(C'_{1}, D'_{3}) → m
31. Segment(D'_{3}, E) → n
32. Segment(C'_{1}, E) → p
33. Angle(E, C'_{1}, D'_{3}) → α
34. OrthogonalLine(C, g) → q
35. Intersect(g, q) → G
36. Midpoint(C, G) → H
37. Dilate(C, If[6 < c ≤ 7, 7 - c, 0], H) → C'_{2}
38. Segment(C, C'_{2}) → r
39. Dilate(H, If[7 < c ≤ 8, 8 - c, 0], G) → H'
40. Segment(H, H') → s
41. Rotate(C, If[8 < c ≤ 9, ((-(c - 8)) * π), (-π)], H) → C'_{3}
42. Rotate(C'_{2}, If[8 < c ≤ 9, ((-(c - 8)) * π), (-π)], H) → I
43. Segment(C'_{3}, I) → r'
44. Midpoint(k) → J
45. Dilate(E, If[9 < c ≤ 10, 10 - c, 0], H) → E'_{1}
46. Dilate(J, If[9 < c ≤ 10, 10 - c, 0], H) → J'
47. Segment(J, J') → t
48. Segment(E, E'_{1}) → f_1
49. Polygon(A, C'_{1}, E, J) → q2, a_2, g_1, e_2, j_1
50. Polygon(E, C, J) → t3, j_2, e_3, c_3
51. Rotate(E, If[11 < c ≤ 12, ((-(c - 11)) * π), (-π)], E) → E'_{2}
52. Rotate(C, If[11 < c ≤ 12, ((-(c - 11)) * π), (-π)], E) → C'_{4}
53. Rotate(J, If[11 < c ≤ 12, ((-(c - 11)) * π), (-π)], E) → J'_{1}
54. Polygon(E'_{2}, C'_{4}, J'_{1}) → t3', j'_{1}, e'_{2}, c'_{4}
55. Segment(J'_{1}, C'_{4}) → h_1
56. Segment(C'_{4}, E) → i_1
57. Segment(E, J'_{1}) → k_1
58. Segment(A, J) → l_1
59. Segment(J, C) → m_1
60. OrthogonalLine(C'_{1}, g) → n_1
61. Intersect(n_1, p_1) → K
62. OrthogonalLine(J'_{1}, g) → q_1
63. Intersect(q_1, g) → L
64. Dilate(J'_{1}, If[13 < c ≤ 14, 14 - c, 0], L) → M
65. Segment(J'_{1}, M) → r_1
66. Dilate(C'_{1}, If[13 < c ≤ 14, 14 - c, 0], K) → N
67. Segment(C'_{1}, N) → s_1
68. Polygon(A, L, J'_{1}, J) → q3, a_3, l_2, t_1, j_3
69. Polygon(L, C'_{1}, J'_{1}) → t4, f_2, l_3, g_2
70. Dilate(J'_{1}, If[15 < c ≤ 16, 16 - c, 0], J) → O
71. Vector(J'_{1}, O) → u
72. Translate(L, u) → L'
73. Translate(C'_{1}, u) → P
74. Translate(J'_{1}, u) → Q
75. Polygon(L', P, Q) → t4', q_2, l', p_2
76. Segment(P, O) → h_2
77. Segment(O, L') → i_2
78. Polygon(A, C'_{1}, K, J) → q4, a_4, k_2, k_3, j_4
79. Polygon(C'_{1}, J'_{1}, K) → t5, k_4, m_2, n_2
80. Translate(C'_{1}, u) → R
81. Translate(J'_{1}, u) → S
82. Translate(K, u) → K'
83. Polygon(R, S, K') → t5', k', r_2, s_2
84. Segment(R, Q) → t_2
85. Segment(R, K') → b_2