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Angle Relationship in a Square

Angle Relationship in a Square
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2017-03-19 11:44
CirclePolygonAngleTransformationGeometric Construction
This example illustrates the angle relationship within a square involving three quarter-circles and several segments. Key mathematical concepts include angles and circles. It is constructed using GeoGebra commands such as Polygon, Segment, Rotate, Circle, etc. Suitable for middle school and above students to learn geometric construction and angle relationships.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Segment(D, A) → j
  3. Rotate(C, ((-π)) / 3, B) → C'_2
  4. Segment(B, C'_2) → l_2
  5. Circle(C'_2, (0.1 * j)) → d_1
  6. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A'_1
  7. Rotate(A'_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 3), ((-π)) / 3], B) → P
  8. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B) → A'
  9. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B) → A''
  10. Rotate(A'', If[13 < e ≤ 14, ((-(e - 13)) * π / 3), ((-π)) / 3], B) → A'''_1
  11. Segment(P, A'''_1) → f''_1
  12. Rotate(A'', If[14 < e ≤ 15, ((e - 14) * π / 3), π / 3], C) → A'''_2
  13. Intersect(d_1, l_2, 1) → F_1
  14. Rotate(F_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 3), ((-π)) / 3], A'''_2) → G_1
  15. Midpoint(B, D) → H_1
  16. Midpoint(H_1, A'''_2) → R_1
  17. Circle(A, B) → e_1
  18. Segment(A, A') → f
  19. Circle(A, Radius[e_1] - f / 6) → p_1
  20. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  21. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  22. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  23. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  24. Segment(R', S) → f'''
  25. Point(f''') → Z
  26. Circle(Z, f / 6) → k_1
  27. OrthogonalLine(Z, f''') → g_1
  28. Intersect(k_1, g_1, 1) → A_1
  29. Point(f''') → W
  30. Circle(W, f / 6) → h_1
  31. OrthogonalLine(W, f''') → f_1
  32. Intersect(h_1, f_1, 1) → B_1
  33. Line(A_1, B_1) → j_1
  34. Segment(A'_1, A'') → f'
  35. Segment(A, B) → g
  36. Segment(B, C) → h
  37. Segment(C, D) → i
  38. Circle(B, (0.1 * j)) → p
  39. Circle(C, (0.15 * j)) → k
  40. Circle(D, (0.15 * j)) → r
  41. Segment(A, C) → l
  42. Segment(D, B) → m
  43. Intersect(l, m) → E
  44. Intersect(r, j, 1) → F
  45. Intersect(r, m, 1) → G
  46. Intersect(r, i, 1) → H
  47. Intersect(k, i, 1) → I
  48. Intersect(k, l, 1) → J
  49. Intersect(k, h, 1) → K
  50. Intersect(p, h, 1) → L
  51. Intersect(p, m, 1) → M
  52. Intersect(p, g, 1) → N
  53. Segment(R, A''') → f''
  54. CircleSector(B, L, N) → s
  55. Dilate(B, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  56. Vector(B, B') → u
  57. Translate(s, u) → s'
  58. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  59. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  60. Vector(C, C') → v
  61. Translate(s'', v) → s'''
  62. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  63. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  64. Vector(D, D') → w
  65. Translate(t, w) → t'
  66. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  67. Dilate(A, If[10 < e ≤ 11, 11 - e, 0], E) → A'_2
  68. Dilate(E, If[11 < e ≤ 12, 12 - e, 0], C) → E'
  69. Dilate(D, If[12 < e ≤ 13, 13 - e, 0], E) → D'_1
  70. Dilate(E, If[13 < e ≤ 14, 14 - e, 0], B) → E'_1
  71. Dilate(D, If[17 < e ≤ 18, 18 - e, 0], E) → D'_2
  72. Vector(D, D'_2) → u_1
  73. Rotate(E, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5707963267948966], E) → E'_2
  74. Rotate(E'_1, If[14 < e ≤ 15, ((e - 14) * π / 2), 1.5708], E) → T
  75. Rotate(E'_2, If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5707963267948966], E) → U
  76. Rotate(T, If[15 < e ≤ 16, ((e - 15) * π / 2), 1.5708], E) → T'
  77. Rotate(U, If[17 < e ≤ 18, ((e - 17) * π / 2), 1.5707963267948966], E) → U'
  78. Rotate(T', If[16 < e ≤ 17, ((e - 16) * π / 2), 1.5708], E) → T''
  79. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  80. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  81. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  82. Vector(B, B'_1) → v_1
  83. Vector(B, B'_3) → w_1
  84. Vector(B, B'_2) → i_1
  85. Circle(B, C) → c_1
  86. Intersect(e_1, j_1, 1) → C_1
  87. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_3
  88. Rotate(A, If[10 < e ≤ 11, ((-(e - 10)) * π / 2), ((-π)) / 2], B) → A'_4
  89. CircleArc(B, A'_4, A) → r_1
  90. Rotate(B, If[11 < e ≤ 12, ((e - 11) * π / 2), π / 2], A) → B'_4
  91. CircleArc(A, B, B'_4) → s_1
  92. Segment(A, B) → n_1
  93. Circle(A, 1.6411491042356954) → t_1
  94. Rotate(B, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], C) → B'_5
  95. CircleArc(C, B'_5, B) → q
  96. Intersect(q, r_1) → O, , ,
  97. Rotate(A'_1, If[14 < e ≤ 15, ((e - 14) * π / 3), π / 3], C) → Q
  98. Segment(Q, A'''_2) → f''_2
  99. Polygon(B, A'''_2, Q) → poly2, q_1, b_1, a'''_2
  100. Intersect(p, f') → V,
  101. Intersect(p, f''_1) → D_1,
  102. Rotate(V, If[15 < e ≤ 16, ((-(e - 15)) * π / 3), ((-π)) / 3], B) → V'
  103. Intersect(d_1, f''_2, 1) → E_1
  104. Dilate(B, If[16 < e ≤ 17, 17 - e, 0], H_1) → B'_6
  105. Segment(B, B'_6) → n
  106. Rotate(B, If[17 < e ≤ 18, ((e - 17) * π), π], H_1) → B'_7
  107. Rotate(B'_6, If[17 < e ≤ 18, ((e - 17) * π), π], H_1) → I_1
  108. Segment(B'_7, I_1) → n'
  109. Intersect(r_1, s_1, 1) → J_1
  110. Segment(B, J_1) → l_1
  111. Segment(Q, O) → m_1
  112. Intersect(l_1, m_1) → K_1
  113. Segment(K_1, H_1) → a_1
  114. Line(A'''_2, a_1) → f_2
  115. Line(A'''_2, n) → g_2
  116. Point(g_2) → L_1
  117. Point(f_2) → M_1
  118. Dilate(B, If[18 < e ≤ 19, 19 - e, 0], J_1) → B'_8
  119. Segment(B, B'_8) → h_2
  120. Dilate(C, If[19 < e ≤ 20, 20 - e, 0], K_1) → C'_1
  121. Segment(Q, C'_1) → i_2
  122. Dilate(K_1, If[20 < e ≤ 21, 21 - e, 0], O) → N_1
  123. Segment(K_1, N_1) → j_2
  124. Dilate(K_1, If[21 < e ≤ 22, 22 - e, 0], H_1) → O_1
  125. Segment(K_1, O_1) → k_2
  126. Dilate(A'''_2, If[22 < e ≤ 23, 23 - e, 0], H_1) → P_1
  127. Vector(A'''_2, P_1) → u_2
  128. CircleSector(C'_2, G_1, F_1) → c_2
  129. Translate(c_2, u_2) → d_2
  130. Angle(B, A'''_2, M_1) → α
  131. Rotate(d_2, If[23 < e ≤ 24, (((e - 23) * 5) * π / 4), (5 * π / 4)], H_1) → e_2
  132. Dilate(A'''_2, If[24 < e ≤ 25, 25 - e, 0], R_1) → Q_1
  133. Vector(A'''_2, Q_1) → v_2
  134. Translate(c_2, v_2) → p_2
  135. Rotate(p_2, If[25 < e ≤ 26, ((e - 25) * ((5 * π / 4) - π / 3)), (5 * π / 4) - π / 3], R_1) → q_2
  136. Dilate(R_1, If[26 < e ≤ 27, 27 - e, 0], H_1) → S_1
  137. Vector(R_1, S_1) → w_2
  138. Translate(q_2, w_2) → r_2
  139. Segment(C'_2, C) → m_2