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V. Pantaloni Chord Problem: Dynamic Geometry

V. Pantaloni Chord Problem: Dynamic Geometry
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2019-08-30 09:23
CirclePolygonAngleTransformationGeometric Construction
This dynamic GeoGebra example illustrates and allows modification of a geometry problem posted by Vincent Pantaloni on Twitter, involving circles, chords, polygons, and various transformations. Core mathematical concepts include properties of circles, relationships between chords and angles, similarity transformations, translation, rotation, reflection, and measurement of distances and angles. The construction uses GeoGebra commands such as Circle, Point, Segment, OrthogonalLine, Intersect, Polygon, Radius, Line, Mirror, Angle, Distance, Midpoint, Min, Vector, Dilate, Translate, Ray, and Rotate, creating multiple points, segments, polygons, and auxiliary lines that can be dynamically adjusted via sliders or dragging. Suitable for high school and above, this resource helps students and teachers explore geometric problems, understand transformations and similarity, and serves as material for mathematical modeling or competition preparation.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Point(c) → D
  4. Segment(C, D) → f
  5. Point(f) → E
  6. OrthogonalLine(E, f) → g
  7. Intersect(c, g, 2) → F
  8. Intersect(c, g, 1) → G
  9. Polygon(D, G, 4) → poly1, h, i, j, k, H, I
  10. Polygon(C, F, 4) → poly2, l, m, n, p, J, K
  11. Radius(c) → a
  12. Point(c) → L
  13. Line(A, L) → q
  14. OrthogonalLine(L, q) → r
  15. Circle(L, Distance[A, L]) → e
  16. Intersect(e, r, 1) → M
  17. Intersect(e, r, 2) → N
  18. OrthogonalLine(M, r) → s
  19. OrthogonalLine(N, r) → t
  20. Intersect(e, q, 2) → O
  21. OrthogonalLine(O, q) → f_1
  22. Mirror(N, O) → N'
  23. Mirror(M, O) → M'
  24. Intersect(s, f_1) → P
  25. Intersect(f_1, t) → Q
  26. Line(N', M') → g_1
  27. Intersect(q, g_1) → R
  28. Angle(D, E, F) → α
  29. Segment(A, L) → h_1
  30. Segment(L, O) → i_1
  31. Segment(M, P) → j_1
  32. Segment(P, O) → k_1
  33. Segment(O, Q) → l_1
  34. Segment(Q, N) → m_1
  35. Segment(Q, M') → n_1
  36. Segment(M', R) → p_1
  37. Segment(R, O) → q_1
  38. Segment(P, N') → r_1
  39. Segment(N', R) → s_1
  40. Segment(M, L) → t_1
  41. Segment(L, N) → a_1
  42. Distance(J, F) → o
  43. Distance(D, G) → u
  44. Segment(F, G) → b_1
  45. Line(D, J) → e_1
  46. Midpoint(J, C) → T
  47. Min({poly1, poly2}) → o_1
  48. Midpoint(D, H) → U
  49. Vector(T, O) → w
  50. Distance(D, I) → d_1
  51. Distance(A, B) → c_1
  52. Line(F, J) → f_2
  53. Circle(C, (2 * c_1)) → c_2
  54. Intersect(c_2, f_2, 2) → S
  55. Segment(J, K) → h_2
  56. Segment(K, C) → m_2
  57. Segment(C, F) → n_2
  58. Dilate(F, If[0 < v ≤ 1, 1 - v, 0], S) → F'_1
  59. Vector(F, F'_1) → u_1
  60. Translate(J, u_1) → S'
  61. Polygon(K, C, F'_1, S') → q1, k_2, c_3, f'_1, s'
  62. Ray(F, J) → i_2
  63. Line(S', K) → j_2
  64. Line(C, K) → p_2
  65. Distance(K, S') → v_1
  66. Line(C, S) → g_2
  67. OrthogonalLine(S', g_2) → l_2
  68. Intersect(g_2, l_2) → V
  69. Dilate(S, If[2 < v ≤ 3, 3 - v, 0], V) → S'_1
  70. Vector(S, S'_1) → u_2
  71. Translate(C, u_2) → C'
  72. Polygon(S', K, C', S'_1) → q2, q_2, r_2, s_2, t_2
  73. Segment(J, F) → a_2
  74. Dilate(C', If[4 < v ≤ 5, 5 - v, 0], N) → C''
  75. Vector(C', C'') → v_2
  76. Translate(S', v_2) → S''
  77. Translate(K, v_2) → K'
  78. Translate(C', v_2) → C''_1
  79. Translate(S'_1, v_2) → W
  80. Polygon(S'', K', C''_1, W) → q2', s'', k', c''_1, w_2
  81. Line(C', K) → b_2
  82. Line(N, b_2) → d_2
  83. Point(d_2) → Z
  84. Angle(Z, N, Q) → β
  85. Rotate(S'', If[5 < v ≤ 6, ((v - 5) * β), β], N) → S'''
  86. Rotate(K', If[5 < v ≤ 6, ((v - 5) * β), β], N) → K''
  87. Rotate(C''_1, If[5 < v ≤ 6, ((v - 5) * β), β], N) → A_1
  88. Rotate(W, If[5 < v ≤ 6, ((v - 5) * β), β], N) → W'
  89. Polygon(S''', K'', A_1, W') → q2'', s''', k'', e_2, w'
  90. Dilate(S''', If[6 < v ≤ 7, 7 - v, 0], N') → B_1
  91. Vector(S''', B_1) → u_3
  92. Translate(S''', u_3) → C_1
  93. Translate(K'', u_3) → K'''
  94. Translate(A_1, u_3) → D_1
  95. Translate(W', u_3) → W''
  96. Polygon(C_1, K''', D_1, W'') → q2''', f_3, k''', g_3, w''
  97. Segment(D, I) → h_3
  98. Segment(I, H) → i_3
  99. Segment(H, G) → j_3
  100. Segment(G, D) → k_3
  101. Line(D, I) → l_3
  102. Circle(G, (2 * c_1)) → d_3
  103. Intersect(d_3, l_3) → E_1, F_2
  104. Dilate(D, If[7 < v ≤ 8, 8 - v, 0], E_1) → D'
  105. Vector(D, D') → v_3
  106. Translate(I, v_3) → I'
  107. Polygon(G, H, I', D') → q3, g_4, h_4, i', d'
  108. Line(I', H) → m_3
  109. OrthogonalLine(G, m_3) → n_3
  110. Intersect(m_3, n_3) → F_1
  111. Dilate(H, If[9 < v ≤ 10, 10 - v, 0], F_1) → H'
  112. Vector(H, H') → w_3
  113. Translate(I', w_3) → I''
  114. Polygon(E_1, I'', H', G) → q4, p_3, i'', h', g_5
  115. Dilate(G, If[11 < v ≤ 12, 12 - v, 0], N) → G'
  116. Vector(G, G') → a_3
  117. Translate(E_1, a_3) → G_1
  118. Translate(I'', a_3) → I'''
  119. Translate(H', a_3) → H''
  120. Translate(G, a_3) → G'_1
  121. Polygon(G_1, I''', H'', G'_1) → q4', q_3, i''', h'', g'_1
  122. Line(G_1, N) → r_3
  123. Angle(G_1, N, M) → γ
  124. Rotate(G_1, If[12 < v ≤ 13, ((v - 12) * γ), γ], N) → H_1
  125. Rotate(I''', If[12 < v ≤ 13, ((v - 12) * γ), γ], N) → I_1
  126. Rotate(H'', If[12 < v ≤ 13, ((v - 12) * γ), γ], N) → H'''
  127. Rotate(G'_1, If[12 < v ≤ 13, ((v - 12) * γ), γ], N) → J_1
  128. Polygon(H_1, I_1, H''', J_1) → q4'', s_3, t_3, h''', b_3
  129. Dilate(N, If[13 < v ≤ 14, 14 - v, 0], D_1) → N'_1
  130. Vector(N, N'_1) → e_3
  131. Translate(H_1, e_3) → K_1
  132. Translate(I_1, e_3) → L_1
  133. Translate(H''', e_3) → M_1
  134. Translate(J_1, e_3) → N_1
  135. Polygon(K_1, L_1, M_1, N_1) → q4''', f_4, i_4, j_4, k_4
  136. Angle(P, M, L) → δ
  137. Angle(R, N', P) → ε
  138. Angle(Q, M', R) → ζ
  139. Line(C, K) → m_4
  140. Line(F, J) → l_4
  141. Line(K, S') → p_4
  142. Line(C, S'_1) → n_4
  143. Line(G, H) → q_4
  144. Line(I', F_1) → r_4
  145. Line(E_1, G) → s_4

AI Drawing Prompt Reference

Draw a dynamic geometry illustration featuring two circles and their intersection points, forming two quadrilaterals. One circle's diameter is a segment, and the other circle's diameter is perpendicular to this segment. Construct two quadrilaterals from these points and lines, highlighting their similarity and symmetry. Include auxiliary lines, angle and distance measurements, and transformations like rotation, reflection, and translation to demonstrate geometric properties.
The prompt above was compiled from the original drawing commands and is for reference only.