动态几何演示：均值不等式与双曲线

GeoGebra 命令及参数

1. Ray((0, 0), (1, 0)) → h_1
2. Point(h_1) → B
3. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
4. Midpoint(A, C) → E
5. Midpoint(A, B) → F
6. Segment(A, C) → j
7. Midpoint(A, E) → G
8. Midpoint(E, C) → H
9. Midpoint(h) → I
10. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], B) → A'
11. Segment(A, A') → k
12. Rotate(A, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], B) → A'_1
13. Rotate(A', If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], B) → A''
14. Segment(A'_1, A'') → k'
15. Rotate(A'_1, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], C) → J
16. Rotate(A'', If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], C) → A'''
17. Segment(J, A''') → k''
18. Rotate(J, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → J'
19. Rotate(A''', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], D) → K
20. Segment(J', K) → k'''
21. Circle(B, (0.15 * Distance[A, B])) → c
22. Segment(A, B) → l
23. Intersect(c, l) → L,
24. Rotate(L, If[4 < a ≤ 5, ((-(a - 4)) * π / 2), ((-π)) / 2], B) → L'
25. CircleSector(B, L', L) → d
26. Dilate(B, If[6 < a ≤ 7, 7 - a, 0], C) → B'
27. Vector(B, B') → u
28. Translate(d, u) → d'
29. Rotate(d', If[6 < a ≤ 7, ((a - 6) * π / 2), π / 2], B') → d''
30. Dilate(C, If[7.5 < a ≤ 8.5, 8.5 - a, 0], D) → C'
31. Vector(C, C') → v
32. Translate(d'', v) → d'''
33. Rotate(d''', If[7.5 < a ≤ 8.5, ((a - 7.5) * π / 2), π / 2], C') → e
34. Dilate(D, If[9 < a ≤ 10, 10 - a, 0], A) → D'
35. Vector(D, D') → w
36. Translate(e, w) → e'
37. Rotate(e', If[9 < a ≤ 10, ((a - 9) * π / 2), π / 2], D') → e''
38. Dilate(A, If[10.5 < a ≤ 11.5, 11.5 - a, 0], B) → A'_2
39. Vector(A, A'_2) → b
40. Translate(e'', b) → e'''
41. Rotate(e''', If[10.5 < a ≤ 11.5, ((a - 10.5) * π / 2), π / 2], A'_2) → p
42. Dilate(A, If[12 < a ≤ 13, 13 - a, 0], G) → A'_3
43. Segment(A, A'_3) → m
44. Dilate(G, If[13 < a ≤ 14, 14 - a, 0], E) → G'
45. Segment(G, G') → n
46. Dilate(E, If[14 < a ≤ 15, 15 - a, 0], H) → E'
47. Segment(E, E') → q
48. Dilate(H, If[15 < a ≤ 16, 16 - a, 0], C) → H'
49. Segment(H, H') → r
50. Dilate(C, If[16 < a ≤ 17, 17 - a, 0], I) → C'_1
51. Segment(C, C'_1) → s
52. Rotate(C, If[17 < a < 18, ((-(a - 17)) * π), (-π)], I) → C'_2
53. Rotate(C'_1, If[17 < a < 18, ((-(a - 17)) * π), (-π)], I) → M
54. Segment(C'_2, M) → s'
55. Rotate(A, If[18 < a < 19, ((a - 18) * π), π], G) → A'_4
56. Rotate(A'_3, If[18 < a < 19, ((a - 18) * π), π], G) → N
57. Segment(A'_4, N) → m'
58. Rotate(A'_4, If[19 < a < 20, ((a - 19) * π), π], E) → O
59. Rotate(N, If[19 < a < 20, ((a - 19) * π), π], E) → N'
60. Segment(O, N') → m''
61. Rotate(O, If[20 < a < 21, ((a - 20) * π), π], H) → O'
62. Rotate(N', If[20 < a < 21, ((a - 20) * π), π], H) → N''
63. Segment(O', N'') → m'''
64. Dilate(B, If[22 < a ≤ 23, 23 - a, 0], G) → B'_1
65. Dilate(I, If[22 < a ≤ 23, 23 - a, 0], G) → I'
66. Segment(I, I') → t
67. Segment(B, B'_1) → f_1
68. Line(B, t) → g_1
69. Point(g_1) → P
70. Angle(P, B, C) → α
71. Dilate(B, If[24 < a ≤ 25, 25 - a, 0], G) → B'_2
72. Vector(B, B'_2) → u_1
73. Translate(p, u_1) → p'
74. Rotate(p', If[24 < a ≤ 25, ((-(a - 24)) * (α + π / 2)), (-(α + π / 2))], B'_2) → p''