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Square and Quarter Circle Action = GoGeometry Action 81!

Square and Quarter Circle Action = GoGeometry Action 81!
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2017-07-28 07:49
CirclePolygonTransformationGeometric ConstructionSlider Animation
This example demonstrates dynamic and modifiable geometric relationships that arise when a quarter circle (sector) is drawn inside a square. The core mathematical concepts include properties of circles and sectors, relationships between squares and polygons, rotations and dilations, as well as midpoints, perpendicular lines, and angle constructions. The construction uses GeoGebra commands such as Ray, Segment, Circle, Intersect, Point, OrthogonalLine, Line, Polygon, Rotate, Dilate, Midpoint, CircleArc, CircleSector, Vector, Translate, Angle, and Mirror, involving objects like points A, B, C, D, polygon poly1, arcs and sectors, and transformed copies (A', A'', etc.). This is suitable for middle or high school geometry classes or for enthusiasts exploring dynamic geometry, helping to understand inscribed figures, symmetry transformations, and proportional relationships.

GeoGebra Commands & Parameters

  1. Ray(A, B_2) → m_2
  2. Segment(A, B_2) → g
  3. Circle(B_2, g) → q_2
  4. Intersect(q_2, m_2, 2) → J_1
  5. Segment(B_2, J_1) → n_2
  6. Point(n_2) → K_1
  7. Segment(B_2, K_1) → r_2
  8. OrthogonalLine(K_1, r_2) → s_2
  9. Circle(K_1, r_2) → t_2
  10. Intersect(t_2, s_2, 2) → L_1
  11. Line(K_1, L_1) → h_3
  12. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  13. Segment(B_2, C) → h
  14. OrthogonalLine(L_1, h) → f_3
  15. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  16. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  17. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  18. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  19. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  20. Segment(R, A''') → f''
  21. Midpoint(A, C) → V
  22. CircleArc(V, C, D) → g_2
  23. Point(g_2) → B
  24. Segment(C, B) → j_1
  25. Segment(A, B) → n
  26. Intersect(n, f'') → Z
  27. Segment(B, Z) → l_1
  28. Segment(A, A') → f
  29. Segment(A'_1, A'') → f'
  30. Segment(C, D) → i
  31. Circle(B_2, (0.2 * r_2)) → p
  32. Segment(D, A) → j
  33. Circle(C, Radius[p]) → k
  34. Circle(A, (0.1 * j)) → q
  35. Circle(D, (0.1 * j)) → r
  36. Segment(A, C) → l
  37. Segment(D, B_2) → m
  38. Intersect(r, j, 1) → F
  39. Intersect(r, m, 1) → G
  40. Intersect(r, i, 1) → H
  41. Intersect(k, i, 1) → I
  42. Intersect(k, l, 1) → J
  43. Intersect(k, h, 1) → K
  44. Intersect(p, h, 1) → L
  45. Intersect(p, m, 1) → M
  46. Intersect(p, g, 1) → N
  47. Intersect(q, g, 1) → O
  48. Intersect(q, l, 1) → P
  49. Intersect(q, j, 1) → Q
  50. CircleSector(B_2, L, N) → s
  51. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  52. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  53. Segment(R', S) → f'''
  54. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  55. Vector(B_2, B') → u
  56. Translate(s, u) → s'
  57. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  58. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  59. Vector(C, C') → v
  60. Translate(s'', v) → s'''
  61. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  62. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  63. Vector(D, D') → w
  64. Translate(t, w) → t'
  65. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  66. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  67. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  68. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  69. Vector(B_2, B'_1) → v_1
  70. Vector(B_2, B'_3) → w_1
  71. Vector(B_2, B'_2) → i_1
  72. Circle(A, n - j_1) → h_2
  73. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  74. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  75. Vector(B_2, B'_4) → u_1
  76. Translate(s, u_1) → s'_1
  77. Line(B, f') → a_1
  78. Point(a_1) → U
  79. Angle(U, B, C) → α
  80. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  81. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  82. Line(A, B_2) → h_1
  83. Angle(B, A, D) → β
  84. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  85. Mirror(C, D) → C'_1
  86. Rotate(C, ((-π)) / 3, B_2) → C'_2
  87. Segment(B_2, C'_2) → f_1
  88. Segment(C, C'_2) → g_1
  89. Intersect(k, g_1, 1) → E
  90. Intersect(k, h) → T,
  91. CircleSector(C, T, E) → d_1
  92. Rotate(D, 15°, C) → D'_2
  93. Angle(D, C, D'_2) → γ
  94. Ray(C, D'_2) → k_1
  95. Rotate(C, (-(15°)), D) → C'_3
  96. Angle(C'_3, D, C) → δ
  97. Ray(D, C'_3) → m_1
  98. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  99. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  100. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  101. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  102. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  103. Rotate(T, 15°, C) → T'
  104. Angle(T, C, T') → ε
  105. Ray(C, T') → q_1
  106. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  107. Vector(C, C'_8) → u_2
  108. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  109. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  110. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  111. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  112. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  113. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  114. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  115. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  116. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  117. Vector(B_2, Q_1) → c_1
  118. Segment(C, K_1) → e_1
  119. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  120. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  121. Line(A, K_1) → k_3
  122. Intersect(t, f'', 1) → G_2
  123. Rotate(B_2, If[10 < e ≤ 11, ((e - 10) * π / 2), π / 2], A) → W
  124. CircleArc(A, B_2, W) → p_1
  125. Rotate(A, If[10 < e ≤ 11, ((e - 10) * π / 2), π / 2], A) → A'_2
  126. Rotate(A', If[10 < e ≤ 11, ((e - 10) * π / 2), π / 2], A) → A''_1
  127. Segment(A'_2, A''_1) → f'_1
  128. Segment(A, C) → n_1
  129. Segment(B_2, D) → r_1
  130. Intersect(r_1, n_1) → G_1
  131. Intersect(p_1, n_1) → M_1,
  132. Line(B_2, M_1) → s_1
  133. Intersect(s_1, f'') → O_1
  134. Dilate(D, If[11 < e ≤ 12, 12 - e, 0], B_2) → D'_3
  135. Segment(D, D'_3) → t_1
  136. Dilate(A, If[12 < e ≤ 13, 13 - e, 0], G_1) → A'_3
  137. Segment(A, A'_3) → b_1
  138. Dilate(G_1, If[13 < e ≤ 14, 14 - e, 0], M_1) → P_1
  139. Segment(G_1, P_1) → f_2
  140. Dilate(M_1, If[14 < e ≤ 15, 15 - e, 0], C) → R_1
  141. Segment(M_1, R_1) → i_2
  142. Dilate(B_2, If[15 < e ≤ 16, 16 - e, 0], M_1) → S_1
  143. Segment(B_2, S_1) → j_2
  144. Dilate(M_1, If[16 < e ≤ 17, 17 - e, 0], O_1) → T_1
  145. Segment(M_1, T_1) → k_2
  146. Dilate(C, If[16 < e ≤ 17, 17 - e, 0], O_1) → C'_7
  147. Segment(C, C'_7) → l_2
  148. Midpoint(D, O_1) → U_1
  149. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], O_1) → V_1
  150. Vector(M_1, V_1) → v_2
  151. Translate(G_1, v_2) → W_1
  152. Translate(P_1, v_2) → Z_1
  153. Segment(W_1, Z_1) → p_2
  154. Rotate(W_1, If[18 < e ≤ 19, ((-(e - 18)) * π / 4), ((-π)) / 4], O_1) → A_2
  155. Rotate(Z_1, If[18 < e ≤ 19, ((-(e - 18)) * π / 4), ((-π)) / 4], O_1) → C_2
  156. Segment(A_2, C_2) → a_2
  157. Rotate(A_2, If[19 < e ≤ 20, ((-(e - 19)) * π), (-π)], U_1) → D_2
  158. Rotate(C_2, If[19 < e ≤ 20, ((-(e - 19)) * π), (-π)], U_1) → E_2
  159. Segment(D_2, E_2) → b_2
  160. CircleSector(A, B_2, W) → c_2