余弦定理的无字证明：三角形面积与分解的几何可视化

GeoGebra 命令及参数

1. Polygon(A, B, C) → t1, c, a, b
2. Semicircle(A, B) → d
3. Intersect(d, OrthogonalLine[C, c]) → D,
4. Angle(A, C, B) → α
5. Angle(B, A, C) → β
6. Angle(C, B, A) → γ
7. If(t ≤ 1, t, 1) → t_1
8. Segment(A, O) → f
9. Dilate(Translate[A, Vector[A, P]], 1 - t_1 + (c * t_1), P) → A_1
10. Dilate(Translate[B, Vector[A, P]], 1 - t_1 + (c * t_1), P) → B_1
11. Dilate(Translate[C, Vector[A, P]], 1 - t_1 + (c * t_1), P) → C_1
12. Segment(A_1, B_1) → g
13. Segment(B_1, C_1) → h
14. Segment(C_1, A_1) → i
15. Angle(A_1, C_1, B_1) → α_1
16. Angle(B_1, A_1, C_1) → β_1
17. Angle(C_1, B_1, A_1) → γ_1
18. If(t < 1, 0, If[t ≥ 1 ∧ t < 2, t - 1, 1]) → t_2
19. If(t_2 < 0.2, (5 * t_2), 1) → t_{21}
20. If(t_2 < 0.2, 0, If[t_2 ≥ 0.2, ((t_2 - 0.2) * 1.25), 1]) → t_{22}
21. Rotate(C + (a, 0), (-α), C) → B'
22. Polygon(A', B', C) → t2, c_1, a', b'
23. Segment(B', O) → j
24. Dilate(Translate[B', Vector[B', Q]], 1 - t_{22} + (b * t_{22}), Q) → B_2
25. Dilate(Translate[A', Vector[B', Q]], 1 - t_{22} + (b * t_{22}), Q) → A_2
26. Dilate(Translate[C, Vector[B', Q]], 1 - t_{22} + (b * t_{22}), Q) → C_2
27. Segment(A_2, B_2) → k
28. Segment(B_2, C_2) → l
29. Segment(C_2, A_2) → m
30. Angle(B_2, C_2, A_2) → ε
31. Angle(C_2, A_2, B_2) → ζ
32. Angle(A_2, B_2, C_2) → η
33. Rotate(C - (b, 0), α, C) → A''
34. Polygon(A'', B'', C) → t3, c_2, a'', b''
35. If(t < 2, 0, If[t ≥ 2 ∧ t < 3, t - 2, 1]) → t_3
36. If(t_3 < 0.2, (5 * t_3), 1) → t_{31}
37. If(t_3 < 0.2, 0, If[t_3 ≥ 0.2, ((t_3 - 0.2) * 1.25), 1]) → t_{32}
38. Segment(B'', C_1) → n
39. Dilate(Translate[B'', Vector[B'', R]], 1 - t_{32} + (a * t_{32}), R) → B_3
40. Dilate(Translate[A'', Vector[B'', R]], 1 - t_{32} + (a * t_{32}), R) → A_3
41. Dilate(Translate[C, Vector[B'', R]], 1 - t_{32} + (a * t_{32}), R) → C_3
42. Segment(A_3, B_3) → p
43. Segment(B_3, C_3) → q
44. Segment(C_3, A_3) → r
45. Angle(B_3, C_3, A_3) → θ
46. Angle(A_3, B_3, C_3) → ι
47. Angle(C_3, A_3, B_3) → κ
48. If(t < 3, 0, If[3 ≥ 2 ∧ t < 4, t - 3, 1]) → t_4
49. Segment(A_1, ((1 - t_4) * A_1) + (t_4 * (x(A_1), y(C_1)))) → s
50. Segment(B_1, ((1 - t_4) * B_1) + (t_4 * (x(B_1), y(C_1)))) → e
51. Angle(C_2, (x(A_1), y(C_1)), A_1) → λ
52. Angle(B_1, (x(B_1), y(C_1)), C_3) → μ
53. Segment(C_3, (x(B_1), y(C_1))) → f_1
54. Segment((x(A_1), y(C_1)), C_2) → g_1