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Proof Without Words by Steve Phelps (13)

Proof Without Words by Steve Phelps (13)
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2015-02-11 09:00
TriangleVectorCoordinate GeometryGeometric Construction
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This example demonstrates the dynamic relationship between isosceles triangles, median lines, and parallelograms by dragging points and a slider. It involves core concepts like median line properties, isosceles triangle determination, and parallelogram construction. Geogebra commands used include points, circles, polygons, lines, orthogonal lines, intersections, conditions, dilations, vectors, translations, rotations, midpoints, and mirrors. Suitable for high school mathematics education to visually understand geometric theorems and transformations.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(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. Line(H, C) → n
  9. OrthogonalLine(K, n) → p
  10. OrthogonalLine(A, n) → q
  11. OrthogonalLine(J, p) → r
  12. Intersect(q, r) → L
  13. Intersect(p, r) → M
  14. Intersect(n, p) → N
  15. Intersect(n, q) → O
  16. Polygon(G, F, A) → poly4, a_1, g_1, f_1
  17. Polygon(F, E, A) → poly5, a_2, f_2, e_1
  18. Intersect(r, h) → P
  19. Polygon(H, P, 4) → poly7, s, t, b_1, c_1, Q, R
  20. Polygon(E, P, Q) → poly6, q_1, e_2, p_1
  21. Polygon(E, R, Q) → poly8, q_2, e_3, r_1
  22. Polygon(E, I, P) → poly9, p_2, e_4, i_1
  23. Polygon(E, C, R) → poly10, r_2, e_5, c_2
  24. If(0 ≤ o < 1, 1 - o, If[o ≥ 1, 0, 1]) → d1
  25. Dilate(R, d1, N) → R'
  26. Vector(R, R') → u
  27. Translate(poly7, u) → poly7'
  28. If(1 ≤ o < 2, 2 - o, If[o ≥ 2, 0, 1]) → d2
  29. Dilate(E, d2, L) → E'
  30. Vector(E, E') → v
  31. Translate(F, v) → F'
  32. Translate(E, v) → E'_1
  33. Translate(A, v) → A'
  34. Polygon(F', E'_1, A') → poly5', a', f', e'_1
  35. If(2 ≤ o < 3, 3 - o, If[o ≥ 3, 0, 1]) → d3
  36. Dilate(G, d3, N) → G'
  37. Vector(G, G') → w
  38. Translate(G, w) → G'_1
  39. Translate(F, w) → F'_1
  40. Translate(A, w) → A'_1
  41. Polygon(G'_1, F'_1, A'_1) → poly4', a'_1, g'_1, f'_1
  42. If(3 ≤ o < 4, 4 - o, If[o ≥ 4, 0, 1]) → d4
  43. Dilate(E, d4, A) → E'_2
  44. Vector(E, E'_2) → u_1
  45. Translate(E, u_1) → E'_3
  46. Translate(R, u_1) → R'_1
  47. Translate(Q, u_1) → Q'
  48. Polygon(E'_3, R'_1, Q') → poly8', q', e'_3, r'_1
  49. If(3 ≤ o < 4, o - 3, If[o ≥ 4, 1, 0]) → r1
  50. Rotate(E'_3, (-(((r1 * 90))°)), E'_3) → S
  51. Rotate(R'_1, (-(((r1 * 90))°)), E'_3) → T
  52. Rotate(Q', (-(((r1 * 90))°)), E'_3) → Q''
  53. Polygon(S, T, Q'') → poly8'', q'', s_1, t_1
  54. Midpoint(M, C) → U
  55. If(4 ≤ o < 5, o - 4, If[o ≥ 5, 1, 0]) → r2
  56. Rotate(E, (-(((r2 * 180))°)), U) → E'_4
  57. Rotate(C, (-(((r2 * 180))°)), U) → C'
  58. Rotate(R, (-(((r2 * 180))°)), U) → R'_2
  59. Polygon(E'_4, C', R'_2) → poly10', r'_2, e'_4, c'
  60. If(5 ≤ o < 6, 6 - o, If[o ≥ 6, 0, 1]) → d5
  61. Dilate(E, d5, A) → E'_5
  62. Mirror(E, p_1) → E'_6
  63. Mirror(I, p_1) → I'
  64. Mirror(P, p_1) → P'
  65. Polygon(E'_6, I', P') → poly9', p', e'_6, i'
  66. Mirror(E, p_1) → E'_7
  67. Mirror(P, p_1) → P'_1
  68. Mirror(Q, p_1) → Q'_1
  69. Polygon(E'_7, P'_1, Q'_1) → poly6', q'_1, e'_7, p'_1
  70. If(6 ≤ o < 7, 7 - o, If[o ≥ 7, 0, 1]) → d6
  71. Dilate(C, d6, O) → C'_1
  72. Vector(C, C'_1) → v_1
  73. Translate(E'_6, v_1) → V
  74. Translate(I', v_1) → I''
  75. Translate(P', v_1) → P''
  76. Polygon(V, I'', P'') → poly9'', p'', v_2, i''
  77. If(7 ≤ o < 8, 8 - o, If[o ≥ 8, 0, 1]) → d7
  78. Dilate(P'_1, d7, C) → W
  79. If(7 ≤ o < 8, o - 7, If[o ≥ 8, 1, 0]) → r3
  80. Vector(P'_1, W) → w_1
  81. Translate(E'_7, w_1) → Z
  82. Translate(P'_1, w_1) → A_1
  83. Translate(Q'_1, w_1) → B_1
  84. Polygon(Z, A_1, B_1) → poly6'', b_2, z_1, a_3
  85. Rotate(Z, ((r3 * 90))°, A_1) → Z'
  86. Rotate(A_1, ((r3 * 90))°, A_1) → C_1
  87. Rotate(B_1, ((r3 * 90))°, A_1) → D_1
  88. Polygon(Z', C_1, D_1) → poly6''', d_1, z', c_3

AI Drawing Prompt Reference

Place two points on the coordinate axes, draw an arc centered at one point, pick a point on the arc, and connect the three points to form a triangle. Based on this triangle, construct three parallelograms outward, and draw relevant altitudes and perpendiculars. Use intersections of perpendiculars to define new points, forming several small triangles and quadrilaterals. Then, use a slider to control a series of dilations and translations, moving parts of the figure along specific directions, dynamically showing the area relationships between triangles and parallelograms, resulting in a proof-without-words geometric animation.
The prompt above was compiled from the original drawing commands and is for reference only.