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Indian Method for Square Root of Two

Indian Method for Square Root of Two
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2017-02-02 02:38
TriangleVectorGeometric Construction
This example demonstrates an ancient Indian method for approximating the square root of two. It involves geometric constructions and properties of isosceles triangles. By constructing isosceles triangles and using geometric transformations, one can intuitively understand the approximation of the square root of two. Main commands used include Polygon, Dilate, OrthogonalLine, Intersect, Segment, Midpoint, Circle, Ray, Vector, Translate. Suitable for middle school and above students to learn geometric constructions and transformations.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Dilate(C, 2 / 3, D) → C'
  3. Dilate(C, 2 / 3, B) → C'_1
  4. OrthogonalLine(C', h) → j
  5. OrthogonalLine(C'_1, g) → k
  6. Intersect(j, k) → E
  7. Intersect(k, i) → F
  8. Segment(A, B) → l
  9. Intersect(l, j) → G
  10. Segment(C', E) → m
  11. Segment(E, C'_1) → n
  12. Segment(A, F) → p
  13. Segment(F, D) → q
  14. Segment(D, C') → r
  15. Segment(C'_1, B) → s
  16. Midpoint(r) → K
  17. OrthogonalLine(K, r) → g_1
  18. Intersect(g_1, l) → L
  19. Dilate(B, 1 / 18, A) → B'
  20. Dilate(D, 1 / 18, A) → D'
  21. Circle(F, D) → c
  22. Intersect(c, p, 1) → H
  23. OrthogonalLine(H, p) → t
  24. OrthogonalLine(F, q) → a
  25. OrthogonalLine(B', l) → b
  26. OrthogonalLine(D', p) → d
  27. OrthogonalLine(G, l) → e
  28. Intersect(d, e) → I
  29. Intersect(g_1, d) → J
  30. Intersect(b, d) → M
  31. Intersect(t, b) → N
  32. Intersect(a, b) → O
  33. Intersect(r, b) → P
  34. OrthogonalLine(D', p) → f_1
  35. Intersect(f_1, s) → Q
  36. Segment(D', Q) → h_1
  37. Segment(F, O) → i_1
  38. Segment(H, N) → j_1
  39. Segment(M, Q) → k_1
  40. Segment(P, M) → l_1
  41. Segment(B', M) → m_1
  42. Segment(L, G) → n_1
  43. Segment(I, G) → p_1
  44. Segment(J, L) → q_1
  45. Ray(B, C'_1) → r_1
  46. Ray(D, C') → s_1
  47. Intersect(r_1, s_1) → R
  48. Vector(J, L) → u
  49. Vector(N, H) → v
  50. Translate(R, u) → R'
  51. Translate(R, v) → R'_1
  52. Translate(R'_1, Vector[(0, 0)]) → S
  53. Translate(R'_1, Vector[(0, 0)]) → T
  54. Translate(R'_1, Vector[(0, 0)]) → U
  55. OrthogonalLine(R'_1, r) → t_1
  56. OrthogonalLine(R', m) → a_1
  57. Intersect(t_1, a_1) → V
  58. Segment(C', R'_1) → b_1
  59. Segment(C'_1, R') → c_1
  60. Segment(R'_1, V) → d_1
  61. Segment(V, R') → e_1
  62. Dilate(R', 1 / 6, C'_1) → R''
  63. OrthogonalLine(R'', c_1) → f_2
  64. Intersect(f_2, m) → W
  65. Intersect(f_2, c_1) → Z
  66. Vector(C'_1, R'') → w
  67. Polygon(E, W, R'', C'_1) → poly2, e_2, w_1, r'', c'_1
  68. Translate(E, w) → E'
  69. Translate(W, w) → W'
  70. Translate(R'', w) → R'''
  71. Translate(C'_1, w) → A_1
  72. Polygon(E', W', R''', A_1) → poly2', e', w', r''', a_2
  73. Translate(E', w) → E''
  74. Translate(W', w) → W''
  75. Translate(R''', w) → B_1
  76. Translate(A_1, w) → C_1
  77. Polygon(E'', W'', B_1, C_1) → poly2'', e'', w'', b_2, c_2
  78. Translate(E'', w) → E'''
  79. Translate(W'', w) → W'''
  80. Translate(B_1, w) → D_1
  81. Translate(C_1, w) → E_1
  82. Polygon(E''', W''', D_1, E_1) → poly2''', e''', w''', d_2, e_3
  83. Translate(E''', w) → F_1
  84. Translate(W''', w) → G_1
  85. Translate(D_1, w) → H_1
  86. Translate(E_1, w) → I_1
  87. Polygon(F_1, G_1, H_1, I_1) → poly3, f_3, g_2, h_2, i_2
  88. Translate(F_1, u_1) → J_1
  89. Translate(G_1, u_1) → K_1
  90. Translate(H_1, u_1) → L_1
  91. Translate(I_1, u_1) → M_1
  92. Polygon(J_1, K_1, L_1, M_1) → poly3', j_2, k_2, l_2, m_2
  93. Translate(J_1, w) → N_1
  94. Translate(K_1, w) → O_1
  95. Translate(L_1, w) → P_1
  96. Translate(M_1, w) → Q_1
  97. Polygon(N_1, O_1, P_1, Q_1) → poly3'', n_2, o_1, p_2, q_2
  98. Intersect(b_1, d_1) → R_1
  99. Polygon(O_1, V, R_1, C') → poly4, o_2, v_1, r_2, c'
  100. Polygon(D, P, O, F) → poly5, d_3, p_3, o, f_4
  101. Polygon(F, O, N, H) → poly6, f_5, o_3, n_3, h_3
  102. Polygon(H, N, B', A) → poly7, h_4, n_4, b', a_3
  103. Polygon(M, J, L, B') → poly8, m_3, j_3, l_3, b'_1
  104. Polygon(J, I, G, L) → poly9, j_4, i_3, g_3, l_4
  105. Polygon(I, Q, B, G) → poly10, i_4, q_3, b_3, g_4
  106. Segment(O, E) → t_2
  107. Segment(I, E) → k_3
  108. Polygon(M, O, E, I) → poly11, m_4, o_4, e_4, i_5
  109. Polygon(I, E, C'_1, Q) → poly12, i_6, e_5, c'_2, q_4
  110. Polygon(O, P, C', E) → poly13, o_5, p_4, c'_3, e_6