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Dissection Proofs of Pythagoras' Theorem

Dissection Proofs of Pythagoras' Theorem
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2021-11-19 12:13
TrianglePolygonVectorTrigonometryGeometric Construction
This example demonstrates geometric dissection proofs of the Pythagorean Theorem, involving core concepts like the theorem itself, geometric dissection, area calculation, vectors, and transformations. It leverages GeoGebra commands and objects such as Segment, Intersect, Polygon, PointIn, Point, Vector, Line, Translate, and Sequence as outlined in the construction summary. Ideal for high school mathematics education to visually illustrate various proofs of the theorem and enhance understanding and application.

GeoGebra Commands & Parameters

  1. Segment((a, (-a)), (0, (-a))) → l
  2. Intersect(xAxis, yAxis) → O
  3. Segment(O, (0, (-a))) → k
  4. Segment(O, (a, 0)) → j
  5. Polygon({O, O + (a, 0), O + (a, (-a)), O + (0, (-a))}) → q1
  6. PointIn(Polygon[{O + (a / 2, 0), O + (a, 0), O + (a, ((-a)) / 2), O + (a / 2, ((-a)) / 2)}]) → C
  7. Point(Segment[O + (1.25, 0), O + (2.99, 0)]) → D
  8. Vector(O - (0, a), D) → u
  9. Line(C, C + u) → p
  10. Intersect(p, l) → H_{3}
  11. Vector(O + (a, 0) + (0, x(O - D)), O) → v
  12. Line(C - u, C - u + v) → j_8
  13. Intersect(j_8, l) → H_5
  14. Intersect(j_8, k) → H_4
  15. Line(C, C + v) → n
  16. Intersect(n, j) → F_2
  17. Polygon({O, D, D + (0, x(D - O)), O + (0, x(D - O))}) → q2
  18. Polygon({C, C - v, C - v + u, C + u}) → q3
  19. Translate(q2, Vector[(-u) - v]) → q2'
  20. Polygon({(0, 0), (b, 0), (b, a)}) → t1
  21. Segment((a, (-a)), (a, 0)) → m
  22. Intersect(p, j) → E_1
  23. Intersect(n, m) → G_1
  24. Intersect(n, k) → F_1
  25. Intersect(p, m) → E_2
  26. Intersect(n, l) → G_2
  27. Intersect(p, k) → H_2
  28. Polygon(I_1, I_1 + E_1 - C, I_1 + P - C, I_1 + G_1 - C) → q4, i_1, f, g, h
  29. Polygon(I_1, I_1 + E_2 - C, I_1 + G_1 - C) → t2, i, q, r
  30. Polygon(I_1, I_1 + E_2 - C, I_1 + Q - C, I_1 + G_2 - C) → q5, s, t, c, d
  31. Polygon(I_1, I_1 + E_1 - C, I_1 + P - C, I_1 + Q - C, I_1 + G_2 - C) → poly1, e, f_1, g_1, h_1, j_1
  32. Polygon(I_2, I_2 + E_1 - C, I_2 + O - C, I_2 + F_1 - C) → q6, i_2, k_1, l_1, m_1
  33. Polygon(I_2, I_2 + E_1 - C, I_2 + F_2 - C) → t3, n_1, p_1, q_1
  34. Polygon(I_2, I_2 + E_2 - C, I_2 + P - C, I_2 + F_2 - C) → q7, r_1, s_1, t_1, a_1
  35. Polygon(I_2, I_2 + E_2 - C, I_2 + P - C, I_2 + O - C, I_2 + F_1 - C) → poly2, b_1, c_1, d_1, e_1, f_2
  36. Polygon(I_3, I_3 + F_2 - C, I_3 + O - C, I_3 + H_4 - C, I_3 + H_5 - C, I_3 + H_{3} - C) → poly3, g_2, h_2, j_2, k_2, i_3, k_8
  37. Polygon(I_3, I_3 + F_1 - C, I_3 + R - C, I_3 + H_{3} - C) → q8, l_2, m_2, n_2, p_2
  38. Polygon(I_3, I_3 + F_1 - C, I_3 + H_2 - C) → t4, q_2, r_2, s_2
  39. Polygon(I_3, I_3 + F_2 - C, I_3 + O - C, I_3 + H_2 - C) → q9, t_2, a_2, b_2, c_2
  40. Polygon(I_4, I_4 + H_2 - C, I_4 + R - C, I_4 + Q - C, I_4 + G_1 - C) → poly4, i_4, d_2, e_2, f_3, g_3
  41. Polygon(I_4, I_4 + H_2 - C, I_4 + R - C, I_4 + G_2 - C) → q10, h_3, j_3, k_3, l_3
  42. Polygon(I_4, I_4 + H_{3} - C, I_4 + Q - C, I_4 + G_1 - C) → q11, m_3, n_3, p_3, q_3
  43. Polygon(I_4, I_4 + H_{3} - C, I_4 + G_2 - C) → t5, r_3, s_3, t_3
  44. Segment(O, O + (b, 0)) → a_3
  45. Segment(O + (b, b), O + (b, 0)) → b_3
  46. Segment(O + (b, b), O + (0, b)) → c_3
  47. Segment(O, O + (0, b)) → d_3
  48. Line(C + u, C + u + v) → e_3
  49. Line(C + v, C + u + v) → f_4
  50. Intersect(f_4, d_3) → K_1
  51. Intersect(f_4, c_3) → K_2
  52. Intersect(e_3, c_3) → K_3
  53. Intersect(e_3, b_3) → K_4
  54. Intersect(e_3, d_3) → K_{12}
  55. Intersect(f_4, j) → K_{10}
  56. Intersect(p, c_3) → K_{11}
  57. Intersect(n, d_3) → K_8
  58. Intersect(p, b_3) → K_5
  59. Intersect(n, a_3) → K_7
  60. Intersect(p, a_3) → K_6
  61. Intersect(n, b_3) → K_9
  62. Polygon(J_1, J_1 + E - O, J_1 + K_9 - O, J_1 + K_8 - O) → q12, g_4, h_4, j_4, k_4
  63. Polygon(J_1, J_1 + K_7 - O, J_1 + K_8 - O) → t6, l_4, m_4, n_4
  64. Polygon(H_1, H_1 + K_2 - F, H_1 + K_8 - F, H_1 + K_9 - F) → t7, p_4, q_4, r_4, s_4
  65. Polygon(L_1 + K_1 - G, L_1 + K_2 - G, L_1) → t8, t_4, a_4, b_4
  66. Polygon(H_1, H_1 + K_2 - F, H_1 + K_1 - F, H_1 + K_8 - F, H_1 + K_7 - F, H_1 + E - F) → poly6, e_4, f_5, g_5, h_5, i_5, j_5
  67. Polygon(H_1, H_1 + K_2 - F, H_1 + K_1 - F, H_1 + O - F, H_1 + E - F) → poly7, k_5, l_5, m_5, n_5, p_5
  68. Polygon(H_1, H_1 + K_2 - F, H_1 + K_{10} - F, H_1 + E - F) → q13, c_4, d_4, q_5, r_5
  69. Polygon(L_1, L_1 + O - G, L_1 + K_{10} - G, L_1 + K_2 - G) → q14, s_5, t_5, a_5, b_5
  70. Sequence(Sequence[Translate[q1, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l1, {{9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}}
  71. Polygon(L_1, L_1 + F - G, L_1 + K_5 - G, L_1 + K_6 - G, L_1 + O - G) → poly5, c_5, d_5, e_5, f_6, g_6
  72. Polygon(M, M + K_5 - E, M + K_6 - E) → t9, h_6, m_6, i_6
  73. Polygon(H_1, H_1 + G - F, H_1 + K_8 - F, H_1 + K_7 - F, H_1 + K_6 - F, H_1 + K_5 - F) → poly8, j_6, k_6, l_6, n_6, p_6, a_8
  74. Polygon(L_1, L_1 + K_3 - G, L_1 + K_4 - G, L_1 + E - G, L_1 + O - G) → poly9, q_6, r_6, s_6, t_6, a_6
  75. Polygon(H_1, H_1 + K_3 - F, H_1 + K_4 - F) → t10, b_6, c_6, d_6
  76. Polygon(H_1, H_1 + G - F, H_1 + K_{12} - F, H_1 + K_4 - F) → q15, e_6, f_7, g_7, h_7
  77. Polygon(J_1, J_1 + E - O, J_1 + K_4 - O, J_1 + K_{12} - O) → q16, i_7, j_7, k_7, l_7
  78. Polygon(L_1, L_1 + (b, 0), L_1 + (b, (-b)), L_1 + (0, (-b))) → q17, m_7, n_7, p_7, q_7
  79. Polygon(H_1, H_1 + G - F, H_1 + K_8 - F, H_1 + K_9 - F) → q18, r_7, s_7, t_7, a_7
  80. Polygon(H_1, H_1 + K_{11} - F, H_1 + K_6 - F, H_1 + E - F) → q19, b_7, c_7, d_7, e_7
  81. Polygon(L_1, L_1 + K_{11} - G, L_1 + K_6 - G, L_1 + O - G) → q20, f_8, g_8, h_8, i_8
  82. Polygon({I_5}) → poly10
  83. Polygon(I_5, I_5 + H_4 - H, I_5 + H_5 - H) → t11, l_8, m_8, n_8
  84. Polygon(I_3, I_3 + F_2 - C, I_3 + O - C, I_3 + R - C, I_3 + H_{3} - C) → poly11, p_8, q_8, r_8, s_8, t_8
  85. Polygon(H_1, H_1 + G - F, H_1 + K_8 - F, H_1 + K_7 - F, H_1 + E - F) → poly12, b_8, c_8, d_8, e_8, f_9
  86. Polygon(J_1, J_1 + K_1 - O, J_1 + K_2 - O, J_1 + K_3 - O, J_1 + K_4 - O, J_1 + E - O) → poly13, g_9, h_9, i_9, j_9, k_9, l_9
  87. Sequence(Sequence[Translate[q2, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l2, {{4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}, {4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4}}
  88. Sequence(Sequence[Translate[q1, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l1', {{9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}, {9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9}}
  89. Polygon({O, B, B - (0, b), O - (0, b)}) → q21
  90. Polygon({A, B, B + (a, 0), A + (a, 0)}) → q22
  91. Polygon({O, A, Rotate[O, -90°, A], Rotate[A, 90°, O]}) → q23
  92. Sequence(Sequence[Translate[q3, Vector[(i * u) + (j * v)]], i, -10, 10], j, -10, 10) → l3, {{13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}, {13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13}}

AI Drawing Prompt Reference

Draw a coordinate plane with the origin as the intersection of the perpendicular axes. From the origin, construct a square (with symmetric top and bottom corners) of side length a and outline its polygonal boundary. Choose a point along a horizontal segment from the x-axis region as a reference, then use vectors originating at points related to the origin to define directions and create lines that intersect the square’s edges, producing several intersection points. Using these points together with the origin and the chosen horizontal reference point, assemble multiple polygon pieces representing a dissection-style proof of the Pythagorean theorem. Finally, translate one selected set of polygon pieces by a vector so the pieces align into a clear cut-and-reassemble arrangement, resulting in a multi-region tiling/puzzle figure.
The prompt above was compiled from the original drawing commands and is for reference only.