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Pythagorean Theorem Proof Without Words: Drag to Explore

Pythagorean Theorem Proof Without Words: Drag to Explore
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2019-07-18 02:19
TriangleCirclePolygonTransformationGeometric Construction
This interactive GeoGebra sketch provides a 'proof without words' for the Pythagorean theorem through a dynamic, draggable construction. By moving key points, users observe how areas or segment relationships change, leading to the visual insight that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse. Core mathematical concepts include right triangles, the Pythagorean theorem, area conservation, and geometric transformations. The construction uses GeoGebra commands such as Intersect, Point, Ray, Mirror, Semicircle, CircleArc, Segment, Polygon, Midpoint, OrthogonalLine, Vector, Centroid, and Translate to build interconnected triangles, semicircles, and auxiliary polygons. This resource is ideal for middle school and above, supporting inquiry-based learning, classroom demonstrations, or self-study to grasp the geometric essence of the Pythagorean theorem.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Ray(A, B) → a
  4. Point(a) → C
  5. Mirror(C, A) → C'
  6. Semicircle(C, C') → c
  7. Intersect(c, yAxis) → D,
  8. CircleArc(A, D, C) → d
  9. Point(d) → E
  10. Segment(C, E) → b
  11. Segment(E, C') → e
  12. Segment(C', C) → f
  13. Polygon(C', C, 4) → poly1, g, h, i, j, F, G
  14. Polygon(C, E, 4) → poly2, k, l, m, n, H, I
  15. Polygon(E, C', 4) → poly3, p, q, r, s, J, K
  16. Midpoint(E, J) → L
  17. OrthogonalLine(L, i) → t
  18. OrthogonalLine(L, t) → a_1
  19. Intersect(a_1, s) → M
  20. Intersect(t, r) → N
  21. Intersect(a_1, q) → O
  22. Intersect(t, e) → P
  23. Polygon(M, L, N, K) → poly4, m_1, l_1, n_1, k_1
  24. Polygon(L, N, J, O) → poly5, l_2, n_2, j_1, o
  25. Polygon(L, P, C', O) → poly6, l_3, p_1, c', o_1
  26. Polygon(P, L, M, E) → poly7, p_2, l_4, m_2, e_1
  27. Vector(L, C) → u
  28. Vector(L, F) → v
  29. Vector(L, C') → w
  30. Vector(L, G) → u_1
  31. Centroid(poly4) → Q
  32. Centroid(poly5) → R
  33. Centroid(poly6) → S
  34. Centroid(poly7) → T
  35. Translate(T, w) → T'
  36. Translate(S, u) → S'
  37. Translate(R, v) → R'
  38. Translate(Q, u_1) → Q'
  39. Segment(Q, Q') → b_1
  40. Segment(R, R') → c_1
  41. Segment(S, S') → d_1
  42. Segment(T, T') → f_1
  43. Point(b_1) → U
  44. Point(c_1) → V
  45. Point(d_1) → W
  46. Point(f_1) → Z
  47. Vector(Q, U) → v_1
  48. Vector(R, V) → w_1
  49. Vector(S, W) → g_1
  50. Vector(T, Z) → h_1
  51. Translate(L, w_1) → L'
  52. Translate(N, w_1) → N'
  53. Translate(J, w_1) → J'
  54. Translate(O, w_1) → O'
  55. Polygon(L', N', J', O') → poly5', l', n', j', o'
  56. Translate(M, v_1) → M'
  57. Translate(L, v_1) → L'_1
  58. Translate(N, v_1) → N'_1
  59. Translate(K, v_1) → K'
  60. Polygon(M', L'_1, N'_1, K') → poly4', m', l'_1, n'_1, k'
  61. Translate(L, g_1) → L'_2
  62. Translate(P, g_1) → P'
  63. Translate(C', g_1) → C''
  64. Translate(O, g_1) → O'_1
  65. Polygon(L'_2, P', C'', O'_1) → poly6', l'_2, p', c'', o'_1
  66. Translate(P, h_1) → P'_1
  67. Translate(L, h_1) → L'_3
  68. Translate(M, h_1) → M'_1
  69. Translate(E, h_1) → E'
  70. Polygon(P'_1, L'_3, M'_1, E') → poly7', p'_1, l'_3, m'_1, e'
  71. Midpoint(H, C) → A_1
  72. Midpoint(C, G) → B_1
  73. Segment(A_1, B_1) → i_1
  74. Point(i_1) → C_1
  75. Vector(A_1, C_1) → q_1
  76. Translate(poly2, q_1) → poly2'