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Classroom Arcs Construction: Build Circle Arcs via Geometric Transforms

Classroom Arcs Construction: Build Circle Arcs via Geometric Transforms
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2023-03-18 02:04
CircleVectorTrigonometryTransformationGeometric Construction
In collections geometriadinamica.es
This example demonstrates how to construct and manipulate circle arcs (arcs) in a dynamic geometry environment, using circle geometry (centers/endpoints) together with geometric transformations to visualize how an arc segment changes under rotation, reflection, and translation. Key mathematical ideas include arc construction (Arc), circle geometry, and constructing auxiliary elements such as lines, segments, and perpendiculars to determine arc endpoints and constraints. Construction highlights from the provided GeoGebra commands/objects: Midpoint for key reference points; If for conditional choices; Intersect to find intersection points that guide the arc; Segment/Line/OrthogonalLine to build sides and perpendicular relationships; Circle plus CircumcircleArc / CircleArc / Arc to generate the required arcs; and Rotate / Mirror / Translate to reposition arcs under transformations, with Vector optionally indicating direction-related information. Suitable for classroom geometry (middle and high school) and also for introductory college-level geometry/algorithmic construction demonstrations.

GeoGebra Commands & Parameters

  1. Midpoint(C, D) → F
  2. Midpoint(F, C) → R
  3. Midpoint(R, F) → V
  4. If(zz < 0, 10, zz) → j
  5. Segment(V, S1) → g_2
  6. Segment(D, B) → e
  7. Line(F, e) → k
  8. Midpoint(C, D) → E
  9. Segment(A, C) → d
  10. Rotate(C, 359.91000000000014°, E) → C''
  11. CircumcircleArc(C, C'', S1) → g
  12. Segment(C, F) → r
  13. Mirror(D, k) → D'_1
  14. Segment(R, S1) → f
  15. Segment(R, S1) → n_1
  16. OrthogonalLine(S1, g_2) → s_1
  17. Intersect(k, s_1) → S
  18. Segment(S1, S) → t_1
  19. Circle(S1, (Distance[S1, S] * (c - 5) / 3)) → c_2
  20. Intersect(c_2, s_1) → T, U
  21. Segment(S1, T) → a_2
  22. Mirror(C, k) → C'
  23. Mirror(C'', k) → C'''
  24. Mirror(S1, k) → S1'
  25. CircumcircleArc(C', C''', S1') → g'
  26. Mirror(S1, k) → S1'_1
  27. Mirror(T, k) → T'
  28. Segment(S1'_1, T') → b_2
  29. Mirror(S1, k) → S1'_2
  30. Mirror(S, k) → S'
  31. Segment(S1'_2, S') → d_2
  32. Mirror(R, k) → R'
  33. Mirror(S1, k) → S1'_3
  34. Segment(R', S1'_3) → f'
  35. Circle(V, C) → e_2
  36. Mirror(C, k) → C'_3
  37. Mirror(S1, k) → S1'_4
  38. Mirror(V, k) → V'
  39. Midpoint(R, C) → W
  40. Circle(C, V) → h_2
  41. Intersect(e_2, h_2, 2) → Z
  42. Polygon(C, Z, V) → poly1, v, c_3, z
  43. Mirror(C, k) → C'_4
  44. Mirror(Z, k) → Z'
  45. Mirror(V, k) → V'_1
  46. Polygon(C'_4, Z', V'_1) → poly1', v'_1, c'_4, z'
  47. Circle(C, i) → h
  48. Intersect(h, r, 1) → G
  49. Mirror(G, k) → G'
  50. Circle(G, G') → p
  51. Intersect(p, k, 1) → H
  52. Polygon(G, H, G') → poly2, g'_1, g_1, h_1
  53. Rotate(C, Max[-2.09, (((-(j - 2)) * 3.14) * 2 / 9)], G) → C'_1
  54. Rotate(C'_1, (-Min[5.24, ((j - 5) * 5.24 / 3)]), H) → I
  55. Segment(I, H) → l
  56. CircleArc(H, I, C'_1) → q
  57. Segment(C'_1, G) → s
  58. CircleArc(G, C'_1, C) → t
  59. Mirror(G, k) → G'_1
  60. Mirror(C'_1, k) → J
  61. Mirror(C, k) → C'_2
  62. Arc(Mirror[Circle[G, C'_1], k], J, C'_2) → t'
  63. Mirror(C'_1, k) → K
  64. Mirror(G, k) → G'_2
  65. Segment(K, G'_2) → s'
  66. Midpoint(C, D) → L
  67. Segment(C, L) → a_1
  68. Circle(C, m) → c_1
  69. Intersect(c_1, a_1, 1) → M
  70. Line(M, k) → b_1
  71. Circle(M, C) → d_1
  72. Intersect(d_1, b_1, 1) → N
  73. OrthogonalLine(N, k) → e_1
  74. Intersect(k, e_1) → O
  75. Circle(N, O) → f_1
  76. Intersect(f_1, b_1, 1) → P
  77. Intersect(c_1, e_1, 1) → Q
  78. Rotate(C, Min[1.57, ((n - 2) * 1.57 / 2)], Q) → C'_5
  79. Rotate(N, Min[1.57, ((n - 5) * 1.57 / 3)], P) → N'
  80. CircleArc(Q, C, C'_5) → k_1
  81. Segment(Q, C'_5) → i_1
  82. CircleArc(P, N, N') → p_1
  83. Segment(P, N') → j_1
  84. Mirror(P, k) → P'
  85. Mirror(N, k) → N'_1
  86. Mirror(N', k) → N''
  87. Arc(Mirror[Circle[P, N], k], N'_1, N'') → q_1
  88. Mirror(Q, k) → Q'
  89. Mirror(C, k) → C'_6
  90. Mirror(C'_5, k) → A_1
  91. Arc(Mirror[Circle[Q, C], k], C'_6, A_1) → r_1
  92. Mirror(P, k) → P'_1
  93. Mirror(N', k) → N''_1
  94. Segment(P'_1, N''_1) → l_1
  95. Mirror(Q, k) → Q'_1
  96. Mirror(C'_5, k) → B_1
  97. Segment(Q'_1, B_1) → m_1
  98. Segment(T2, T1) → f_2
  99. Mirror(D, f_2) → D'
  100. Mirror(B, f_2) → B'
  101. Segment(D', B') → e'
  102. Mirror(A, f_2) → A'
  103. Mirror(C, f_2) → C'_7
  104. Segment(A', C'_7) → d'
  105. Vector(A, B') → u_1
  106. Translate(pic1, u_1) → pic1'
  107. Translate(pic1, u_1) → pic1'_1
  108. Midpoint(C, D) → D_1
  109. Midpoint(D', C'_7) → E_1
  110. Segment(D', E_1) → i_2
  111. Distance(D', E_1) → distanciaD'E1
  112. Circle(D', Modifica) → k_2
  113. Intersect(k_2, i_2) → G_1, N_1
  114. Mirror(G_1, E_1) → H_1
  115. Polygon(G_1, H_1, 3) → poly3, j_2, l_2, m_2, I_1
  116. Circle(G_1, D') → p_2
  117. Intersect(p_2, m_2) → K_1, O_1
  118. CircleArc(G_1, K_1, D') → q_2
  119. Line(E_1, e') → n_2
  120. Mirror(G_1, n_2) → L_1
  121. Mirror(K_1, n_2) → M_1
  122. Mirror(D', n_2) → D''
  123. Arc(Mirror[Circle[G_1, K_1], n_2], M_1, D'') → r_2
  124. CircleArc(I_1, M_1, K_1) → s_2

AI Drawing Prompt Reference

Draw a series of circle arcs in a geometric figure. These arcs are constructed using circle geometry and auxiliary elements such as line segments, lines, and perpendiculars. The construction process first determines the endpoints and constraints of the arcs, then uses transformations like rotation, reflection, and translation to show how the arcs change in different positions. The figure includes multiple arcs, circles, line segments, and lines, demonstrating the impact of geometric transformations on the shape of the arcs.
The prompt above was compiled from the original drawing commands and is for reference only.