当ABCD为矩形时，为何PQRS为菱形？

GeoGebra 命令及参数

1. Segment(A, B) → a_1
2. Segment(B, C) → b_1
3. Segment(C, D) → c_1
4. Segment(D, A) → d_1
5. Midpoint(A, B) → P
6. Midpoint(B, C) → Q
7. Midpoint(C, D) → R
8. Midpoint(D, A) → S
9. Segment(P, Q) → e_1
10. Segment(Q, R) → f_1
11. Segment(R, S) → g_1
12. Segment(S, P) → h_1
13. Segment(A, C) → i_1
14. Segment(B, D) → j_1
15. Angle(P, S, R) → w_1
16. Angle(R, Q, P) → y_1
17. Angle(Q, P, S) → x_1
18. Angle(S, R, Q) → z_1
19. Line(A, b_1) → l_1
20. Line(C, a_1) → l_2
21. Intersect(l_1, l_2) → D_2
22. Segment(A, D_2) → d_2
23. Segment(D_2, C) → c_2
24. Segment(B, D_2) → j_2
25. Midpoint(C, D_2) → R_2
26. Midpoint(A, D_2) → S_2
27. Segment(Q, R_2) → f_2
28. Segment(R_2, S_2) → g_2
29. Segment(S_2, P) → h_2
30. Angle(P, S_2, R_2) → w_2
31. Angle(S_2, R_2, Q) → z_2
32. Angle(R_2, Q, P) → y_2
33. Angle(Q, P, S_2) → x_2
34. Circle(B, A) → l_3
35. Point(l_3) → C_3
36. Segment(B, C_3) → b_3
37. Circle(C_3, Segment[C_3, B]) → l_4
38. Circle(A, Segment[A, B]) → l_5
39. Intersect(l_4, l_5, 2) → D_3
40. Segment(C_3, D_3) → c_3
41. Segment(A, D_3) → d_3
42. Midpoint(B, C_3) → Q_3
43. Midpoint(C_3, D_3) → R_3
44. Midpoint(D_3, A) → S_3
45. Segment(P, Q_3) → e_3
46. Segment(S_3, R_3) → g_3
47. Segment(S_3, P) → h_3
48. Segment(R_3, Q_3) → f_3
49. Segment(B, D_3) → j_3
50. Segment(A, C_3) → i_3
51. Angle(Q_3, P, S_3) → x_3
52. Angle(R_3, Q_3, P) → y_3
53. Angle(S_3, R_3, Q_3) → z_3
54. Angle(P, S_3, R_3) → w_3
55. OrthogonalLine(B, a_1) → l_6
56. Point(l_6) → C_4
57. Segment(B, C_4) → b_4
58. OrthogonalLine(C_4, b_4) → l_7
59. Line(A, b_4) → l_8
60. Intersect(l_7, l_8) → D_4
61. Segment(A, D_4) → d_4
62. Segment(D_4, C_4) → c_4
63. Midpoint(B, C_4) → Q_4
64. Midpoint(C_4, D_4) → R_4
65. Midpoint(D_4, A) → S_4
66. Segment(P, Q_4) → e_4
67. Segment(S_4, R_4) → g_4
68. Segment(S_4, P) → h_4
69. Segment(R_4, Q_4) → f_4
70. Segment(A, C_4) → i_4
71. Segment(B, D_4) → j_4
72. Angle(P, S_4, R_4) → w_4
73. Angle(S_4, R_4, Q_4) → z_4
74. Angle(R_4, Q_4, P) → y_4
75. Angle(Q_4, P, S_4) → x_4
76. Circle(B, A) → l_9
77. OrthogonalLine(B, a_1) → l_{10}
78. Intersect(l_9, l_{10}, 2) → C_5
79. Segment(B, C_5) → b_5
80. OrthogonalLine(C_5, b_5) → l_{11}
81. OrthogonalLine(A, a_1) → l_{12}
82. Intersect(l_{11}, l_{12}) → D_5
83. Segment(C_5, D_5) → c_5
84. Segment(D_5, A) → d_5
85. Midpoint(B, C_5) → Q_5
86. Midpoint(C_5, D_5) → R_5
87. Midpoint(D_5, A) → S_5
88. Segment(P, Q_5) → e_5
89. Segment(S_5, R_5) → g_5
90. Segment(R_5, Q_5) → f_5
91. Segment(S_5, P) → h_5
92. Segment(A, C_5) → i_5
93. Segment(B, D_5) → j_5
94. Angle(P, S_5, R_5) → w_5
95. Angle(Q_5, P, S_5) → x_5
96. Angle(R_5, Q_5, P) → y_5
97. Angle(S_5, R_5, Q_5) → z_5