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Adrian Kloskowski - 2nd Liu Hui Pythagorean Proof

Adrian Kloskowski - 2nd Liu Hui Pythagorean Proof
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2017-12-02 01:49
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the second proof of the Pythagorean Theorem by Liu Hui, illustrating the theorem's validity through geometric figures and auxiliary lines. Core mathematical concepts include the Pythagorean Theorem, linear equations, cone, sphere, linear programming or linear optimization, median value. Construction highlights involve using commands and objects in GeoGebra such as Segment, Midpoint, OrthogonalLine, Semicircle, Intersect, CircleArc, Point, Polygon, Rotate, Dilate, Vector, Translate, Circle, Mirror. Suitable for high school mathematics learning and geometric construction practice.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Midpoint(f) → C
  3. OrthogonalLine(C, f) → g
  4. Semicircle(A, B) → c
  5. Intersect(c, g) → D,
  6. CircleArc(C, D, A) → d
  7. Point(d) → E
  8. Segment(B, E) → h
  9. Segment(E, A) → i
  10. Polygon(A, E, 4) → poly1, j, k, l, m, F, G
  11. Polygon(E, B, 4) → poly2, n, p, q, r, H, I
  12. Polygon(B, A, 4) → poly3, s, t, a_3, b, J, K
  13. Segment(B, H) → e
  14. Segment(H, I) → f_1
  15. Segment(I, E) → g_1
  16. Segment(E, F) → h_1
  17. Segment(F, G) → i_1
  18. Segment(G, A) → j_1
  19. Segment(A, J) → k_1
  20. Segment(J, K) → l_1
  21. Segment(K, B) → m_1
  22. Segment(F, A) → n_1
  23. Polygon(F, E, A) → t1, a_1, f_2, e_1
  24. Rotate(F, (-(((90 * o))°)), F) → F'
  25. Rotate(E, (-(((90 * o))°)), F) → E'
  26. Rotate(A, (-(((90 * o))°)), F) → A'
  27. Polygon(F', E', A') → t1', a', f', e'
  28. Dilate(B, o, F) → B'
  29. Vector(F, B') → u
  30. Translate(F', u) → F''
  31. Translate(E', u) → E''
  32. Translate(A', u) → A''
  33. Polygon(F'', E'', A'') → t1'', a'', f'', e''
  34. Polygon(F, A, G) → t2, g_2, f_3, a_2
  35. Rotate(F, (-(((90 * o))°)), A) → F'_1
  36. Rotate(A, (-(((90 * o))°)), A) → A'_1
  37. Rotate(G, (-(((90 * o))°)), A) → G'
  38. Polygon(F'_1, A'_1, G') → t2', g', f'_1, a'_1
  39. Dilate(J, o, A) → J'
  40. Vector(A, J') → v
  41. Translate(F'_1, v) → L
  42. Translate(A'_1, v) → M
  43. Translate(G', v) → G''
  44. Polygon(L, M, G'') → t2'', g'', l_2, m_2
  45. Circle(B, Distance[A, B]) → c_1
  46. Intersect(c_1, f_1) → N,
  47. Intersect(c_1, g_1) → O,
  48. OrthogonalLine(N, f_1) → p_1
  49. OrthogonalLine(O, g_1) → q_1
  50. Intersect(q_1, p_1) → P
  51. Segment(N, P) → r_1
  52. Segment(P, O) → s_1
  53. Segment(P, B) → t_1
  54. Segment(N, B) → b_1
  55. Segment(O, B) → d_1
  56. Polygon(N, H, B) → t3, b_2, n_2, h_2
  57. Dilate(A, o, N) → A'_2
  58. Vector(N, A'_2) → w
  59. Translate(N, w) → N'
  60. Translate(H, w) → H'
  61. Translate(B, w) → B'_1
  62. Polygon(N', H', B'_1) → t3', b'_1, n', h'
  63. Translate(N, u_1) → N'_1
  64. Translate(H, u_1) → H'_1
  65. Translate(B, u_1) → B'_2
  66. Polygon(N'_1, H'_1, B'_2) → t3'_1, b'_2, n'_1, h'_1
  67. Polygon(N, B, P) → t4, p_2, n_3, b_3
  68. Rotate(N, ((90 * o))°, B) → N'_2
  69. Rotate(B, ((90 * o))°, B) → B'_3
  70. Rotate(P, ((90 * o))°, B) → P'
  71. Polygon(N'_2, B'_3, P') → t4', p', n'_2, b'_3
  72. Dilate(H', o, I) → H''
  73. Vector(I, H'') → v_1
  74. Polygon(N, P, O, I) → q1, n_4, p_3, o_1, i_2
  75. Translate(N, v_1) → N'_3
  76. Translate(P, v_1) → P'_1
  77. Translate(O, v_1) → O'
  78. Translate(I, v_1) → I'
  79. Polygon(N'_3, P'_1, O', I') → q1', n'_3, p'_1, o', i'
  80. Dilate(A, o, O) → A'_3
  81. Polygon(B, A'_3, E) → t5, e_2, b_4, a'_3
  82. Rotate(B, ((90 * o))°, B) → B'_4
  83. Rotate(A'_3, ((90 * o))°, B) → Q
  84. Rotate(E, ((90 * o))°, B) → E'_1
  85. Polygon(B'_4, Q, E'_1) → t5', e'_1, b'_4, q_2
  86. Mirror(P, e) → P'_2
  87. Mirror(O, e) → O'_1
  88. Segment(P'_2, O'_1) → j_2
  89. Dilate(P'_2, o, P) → R
  90. Dilate(O'_1, o, O) → S
  91. Polygon(B, S, R) → t6, r_2, b_5, s_2
  92. Vector(B, B'_1) → w_1
  93. Translate(B, w_1) → B'_5
  94. Translate(S, w_1) → S'
  95. Translate(R, w_1) → R'
  96. Polygon(B'_5, S', R') → t6', r', b'_5, s'