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Proof Without Words by Steve Phelps (6)

Proof Without Words by Steve Phelps (6)
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2015-02-18 23:22
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This is a geometric construction example demonstrating a proof without words. By dragging points and sliders, observe the changes in the figure to understand the underlying mathematical principles. It constructs geometric objects such as triangles, arcs, and polygons, using GeoGebra commands like points, intersections, vectors, and transformations. Suitable for high school and above students or teachers to help understand geometric transformations and proof methods.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. CircleArc(B, D, A) → c
  4. Point(c) → E
  5. Polygon(A, E, 4) → poly1, a, b, d, e, F, G
  6. Polygon(E, C, 4) → poly2, f, g, h, i, H, I
  7. Polygon(C, A, 4) → poly3, j, k, l, m, J, K
  8. AngularBisector(A, E, C) → n
  9. Intersect(n, j) → L
  10. OrthogonalLine(L, a) → p
  11. OrthogonalLine(L, f) → q
  12. Midpoint(G, E) → M
  13. Midpoint(E, H) → N
  14. Intersect(p, a) → O
  15. Intersect(q, f) → P
  16. Mirror(O, M) → O'
  17. Mirror(P, N) → P'
  18. Line(O, j) → r
  19. Line(P, j) → s
  20. Intersect(r, e) → Q
  21. Intersect(s, g) → R
  22. Mirror(Q, M) → Q'
  23. Mirror(R, N) → R'
  24. Polygon(Q, O, 4) → poly4, t, a_1, b_1, c_1, S, T
  25. Polygon(P, R, 4) → poly5, d_1, e_1, f_1, g_1, U, V
  26. Polygon(Q, G, O') → poly6, o', q_1, g_2
  27. Polygon(O', F, Q') → poly7, q', o'_1, f_2
  28. Polygon(Q', O, E) → poly8, e_2, q'_1, o
  29. Polygon(O, A, Q) → poly9, q_2, o_1, a_2
  30. Polygon(E, P, R') → poly10, r', e_3, p_1
  31. Polygon(P, C, R) → poly11, r_1, p_2, c_2
  32. Polygon(R, P', H) → poly12, h_1, r_2, p'
  33. Polygon(I, R', P') → poly13, p'_1, i_1, r'_1
  34. If(0 ≤ u ≤ 1, 1 - u, If[u > 1, 0, 1]) → d1
  35. Dilate(O', d1, A) → O''
  36. Vector(O', O'') → v
  37. Translate(poly4, v) → poly4'
  38. Dilate(R, d1, K) → R'_1
  39. Vector(R, R'_1) → w
  40. Translate(poly5, w) → poly5'
  41. If(1 ≤ u ≤ 2, 2 - u, If[u > 2, 0, 1]) → d2
  42. Dilate(R, d2, J) → R'_2
  43. Vector(R, R'_2) → u_1
  44. Translate(R, u_1) → R'_3
  45. Translate(P', u_1) → P''
  46. Translate(H, u_1) → H'
  47. Polygon(R'_3, P'', H') → poly12', h', r'_3, p''
  48. Dilate(P, d2, W) → P'_1
  49. Vector(P, P'_1) → v_1
  50. Translate(E, v_1) → E'
  51. Translate(P, v_1) → P'_2
  52. Translate(R', v_1) → R''
  53. Polygon(E', P'_2, R'') → poly10', r'', e', p'_2
  54. Dilate(P, d2, L) → P'_3
  55. Vector(P, P'_3) → w_1
  56. Translate(P, w_1) → P'_4
  57. Translate(C, w_1) → C'
  58. Translate(R, w_1) → R'_4
  59. Polygon(P'_4, C', R'_4) → poly11', r'_4, p'_4, c'
  60. Dilate(R', d2, R'') → R''_1
  61. Vector(R', R''_1) → j_1
  62. Translate(I, j_1) → I'
  63. Translate(R', j_1) → R''_2
  64. Translate(P', j_1) → P''_1
  65. Polygon(I', R''_2, P''_1) → poly13', p''_1, i', r''_2
  66. If(2 ≤ u ≤ 3, 3 - u, If[u > 3, 0, 1]) → d3
  67. Dilate(G, d3, C') → G'
  68. Vector(G, G') → k_1
  69. Translate(Q, k_1) → Q'_1
  70. Translate(G, k_1) → G'_1
  71. Translate(O', k_1) → O''_1
  72. Polygon(Q'_1, G'_1, O''_1) → poly6', o''_1, q'_2, g'_1
  73. Dilate(O, d3, R'') → O'_1
  74. Vector(O, O'_1) → l_1
  75. Translate(Q', l_1) → Q''
  76. Translate(O, l_1) → O'_2
  77. Translate(E, l_1) → E'_1
  78. Polygon(Q'', O'_2, E'_1) → poly8', e'_1, q'', o'_2
  79. Dilate(A, d3, H') → A'
  80. Vector(A, A') → m_1
  81. Dilate(F, d3, E') → F'
  82. Vector(F, F') → n_1
  83. Translate(O, m_1) → O'_3
  84. Translate(A, m_1) → A'_1
  85. Translate(Q, m_1) → Q'_2
  86. Polygon(O'_3, A'_1, Q'_2) → poly9', q'_3, o'_3, a'_1
  87. Translate(O', n_1) → O''_2
  88. Translate(F, n_1) → F'_1
  89. Translate(Q', n_1) → Q''_1
  90. Polygon(O''_2, F'_1, Q''_1) → poly7', q''_1, o''_2, f'_1

AI Drawing Prompt Reference

Draw a geometric construction with a coordinate system, points, arcs, and polygons. Start by creating points on the axes, then use them to form arcs and polygons. Next, create new points and lines using angle bisectors and perpendiculars. Finally, apply reflections and dilations to transform the polygons and combine them, demonstrating geometric transformations and proof methods.
The prompt above was compiled from the original drawing commands and is for reference only.