Why Is It Always a Parallelogram?

GeoGebra Commands & Parameters

1. Line(A, B) → f
2. Point(f) → F
3. Ray(F, B) → k
4. PointIn(a) → C
5. Point(k) → E
6. Angle(E, C, F) → α
7. Midpoint(E, F) → G
8. Ray(C, F) → g
9. Ray(C, E) → h
10. Line(E, g) → i
11. Line(F, h) → j
12. Intersect(i, j) → H
13. Angle(F, H, E) → β
14. Angle(E, C, F) → γ
15. Segment(C, E) → l
16. Segment(C, F) → m
17. Min(l, m) → c
18. Circle(C, (0.25 * c)) → d
19. Intersect(d, h) → D,
20. Intersect(d, g, 1) → I
21. Dilate(E, If[0 < e ≤ 1, 1 - e, 0], G) → E'
22. Segment(E, E') → n
23. Dilate(G, If[1 < e ≤ 2, 2 - e, 0], F) → G'
24. Segment(G, G') → p
25. Rotate(E, If[2 < e ≤ 3, ((e - 2) * π), π], G) → E'_1
26. Rotate(E', If[2 < e ≤ 3, ((e - 2) * π), π], G) → E''
27. Segment(E'_1, E'') → n'
28. Dilate(C, If[3 < e ≤ 4, 4 - e, 0], F) → C'
29. Dilate(C, If[3 < e ≤ 4, 4 - e, 0], E) → C'_1
30. Segment(C, C'_1) → q
31. Segment(C, C') → r
32. Rotate(D, If[4 < e ≤ 5, ((e - 4) * α), α], C) → D'
33. CircleSector(C, D, D') → s
34. Dilate(C, If[5 < e ≤ 6, 6 - e, 0], H) → C'_2
35. Vector(C, C'_2) → u
36. Translate(s, u) → s'
37. Rotate(s', If[6 < e ≤ 7, ((e - 6) * π), π], H) → s''
38. Dilate(H, If[7 < e ≤ 8, 8 - e, 0], F) → H'
39. Dilate(H, If[7 < e ≤ 8, 8 - e, 0], E) → H'_1
40. Segment(H, H'_1) → t
41. Segment(H, H') → f_1
42. Dilate(F, If[8 < e ≤ 9, 9 - e, 0], H) → F'
43. Vector(F, F') → v
44. Translate(C, v) → C'_3
45. Translate(C', v) → C''
46. Segment(C'_3, C'') → r'
47. Dilate(C, If[9 < e ≤ 10, 10 - e, 0], F) → C'_4
48. Vector(C, C'_4) → w
49. Translate(C, w) → C'_5
50. Translate(C'_1, w) → J
51. Segment(C'_5, J) → q'
52. Polygon(C, E, H, F) → poly1, c_1, e_1, h_1, f_2