Mysterious Relationship of Angle Bisectors in Any Concave Quadrilateral

GeoGebra Commands & Parameters

1. Rotate(A, (a * π / 180), B) → A'
2. Angle(A, B, A') → α
3. Rotate(A, α / 2, B) → A'_2
4. Ray(B, A'_2) → r
5. Ray(B, A') → g
6. Point(g) → C
7. Segment(B, C) → h
8. Polygon(B, A, C) → poly1, c, b, a_1
9. PointIn(poly1) → D
10. Segment(C, D) → i
11. Segment(D, A) → j
12. Angle(B, C, D) → β
13. Rotate(B, β / 2, C) → B'
14. Ray(C, B') → l
15. Angle(D, A, B) → γ
16. Rotate(D, γ / 2, A) → D'
17. Ray(A, D') → n
18. Intersect(n, l) → J
19. Segment(C, J) → g_1
20. Segment(J, A) → h_1
21. Segment(A, B) → f
22. Min({f, i, h, j, g_1, h_1}) → d
23. Circle(B, (0.2 * d)) → p
24. Intersect(p, r, 1) → N
25. Circle(B, (2 * Distance[B, N])) → r_1
26. Intersect(r_1, r) → O,
27. Angle(A, D, C) → δ
28. Circle(D, (0.2 * d)) → q
29. Circle(C, (0.2 * d)) → e
30. Circle(A, (0.2 * d)) → k
31. Intersect(e, h, 1) → E
32. Intersect(q, j, 1) → F
33. Intersect(k, j, 1) → G
34. Intersect(p, f, 1) → H
35. Angle(B, C, B') → ε
36. Rotate(A, δ / 2, D) → A'_1
37. Angle(A, D, A'_1) → ζ
38. Angle(D, A, D') → η
39. Angle(A, B, A'_2) → θ
40. Ray(D, A'_1) → m
41. Intersect(k, n) → K,
42. Intersect(q, m, 1) → L
43. Intersect(e, l) → M,
44. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], J) → A'_3
45. Segment(A, A'_3) → s
46. Polygon(B, A, D, C) → poly2, b_1, a_2, d_1, c_1
47. CircleSector(A, G, K) → t
48. Dilate(C, If[1 < o ≤ 2, 2 - o, 0], J) → C'
49. Segment(C, C') → f_1
50. CircleSector(C, E, M) → e_1
51. Rotate(t, If[2 < o ≤ 3, ((o - 2) * η), η], A) → t'
52. Rotate(e_1, If[3 < o ≤ 4, ((o - 3) * ε), ε], C) → k_1
53. Dilate(B, If[4 < o ≤ 5, 5 - o, 0], O) → B'_1
54. Segment(B, B'_1) → i_1
55. Dilate(D, If[5 < o ≤ 6, 6 - o, 0], L) → D'_1
56. Segment(D, D'_1) → j_1
57. CircleSector(B, H, N) → p_1
58. CircleSector(D, F, L) → q_1
59. Ray(D, L) → l_1
60. Rotate(p_1, If[6 < o ≤ 7, ((o - 6) * θ), θ], B) → s_1
61. Rotate(q_1, If[7 < o ≤ 8, ((o - 7) * ζ), ζ], D) → t_1
62. Line(J, i_1) → m_1
63. Circle(J, 3) → c_2
64. Intersect(c_2, m_1, 2) → P
65. Angle(A, J, P) → κ
66. Dilate(B, If[8 < o ≤ 9, 9 - o, 0], J) → B'_2
67. Vector(B, B'_2) → u
68. Translate(s_1, u) → d_2
69. Rotate(d_2, If[9 < o ≤ 10, ((-(o - 9)) * κ), (-κ)], J) → e_2
70. Midpoint(J, C) → Q
71. Dilate(D, If[10 < o ≤ 11, 11 - o, 0], Q) → D'_2
72. Vector(D, D'_2) → v
73. Translate(t_1, v) → f_2
74. Dilate(Q, If[11 < o ≤ 12, 12 - o, 0], J) → Q'
75. Rotate(A, θ, J) → A'_4
76. Rotate(A'_3, θ, J) → R
77. Segment(A'_4, R) → s'
78. Line(Q', s') → n_1
79. Vector(Q, Q') → w
80. Translate(f_2, w) → g_2
81. Line(J, l_1) → h_2
82. Circle(J, 3) → k_2
83. Intersect(k_2, h_2) → S, T
84. Angle(A'_4, J, T) → ρ
85. Rotate(g_2, If[11 < o ≤ 12, ((-(o - 11)) * ρ), (-ρ)], Q') → p_2