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Pythagorean Theorem Dissection Proof via Translation

Pythagorean Theorem Dissection Proof via Translation
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2015-07-27 09:27
TrianglePolygonVectorTransformationGeometric Construction
This GeoGebra example demonstrates a dissection proof of the Pythagorean theorem by translating, rotating, and reflecting pieces of a right triangle and related polygons to show that the sum of the squares on the legs equals the square on the hypotenuse. It covers core concepts such as the Pythagorean theorem, area conservation, triangle and polygon dissection, vectors, translation, symmetry, rotation, and geometric invariants. The construction uses Segment, Point, Angle, Polygon, Midpoint, Vector, Translate, Dilate, Circle, Semicircle, Ray, Intersect, and Line to create objects a, b, c, poly2, poly3, and transformed points D', C', E', and B'. It is suitable for middle-school to high-school geometry lessons, mathematical experiments, and GeoGebra-based theorem visualization.

GeoGebra Commands & Parameters

  1. Segment(A, B) → c
  2. Semicircle(A, B) → c_2
  3. Point(c_2) → C
  4. Segment(B, C) → b
  5. Segment(C, A) → a
  6. Angle(A, C, B) → α
  7. Polygon(C, B, 4) → small_1, e, f, g, h, D, E
  8. Polygon(A, C, 4) → poly2, i, j, k, l, F, G
  9. Polygon(B, A, 4) → poly3, m, n, p, q, H, I
  10. Midpoint(D, C) → J
  11. Midpoint(F, A) → L
  12. Midpoint(I, A) → K
  13. Vector(J, K) → v
  14. Translate(D, v) → D'
  15. Translate(C, v) → C'
  16. Translate(E, v) → E'
  17. Translate(B, v) → B'
  18. Ray(B', D') → r
  19. Ray(D', E') → s
  20. Ray(E', C') → t
  21. Ray(C', B') → a_1
  22. Segment(I, H) → b_1
  23. Segment(H, A) → c_1
  24. Segment(B, I) → d_1
  25. Intersect(a_1, b_1) → M
  26. Intersect(t, c_1) → N
  27. Intersect(r, d_1) → O
  28. Intersect(c, s) → P
  29. Dilate(D', 1 - equals / 100, A) → D''
  30. Circle(G, Segment[B', M]) → e_1
  31. Segment(F, G) → g_1
  32. Segment(A, G) → h_1
  33. Segment(C, F) → i_1
  34. Intersect(e_1, h_1, 1) → Q
  35. Circle(A, Segment[G, Q]) → k_1
  36. Circle(C, Segment[G, Q]) → q_1
  37. Circle(F, Segment[G, Q]) → r_1
  38. Intersect(e_1, g_1, 1) → R
  39. Intersect(r_1, g_1, 1) → S
  40. Intersect(q_1, a, 1) → T
  41. Intersect(k_1, a, 1) → U
  42. Intersect(k_1, h_1, 1) → V
  43. Intersect(q_1, i_1, 1) → W
  44. Intersect(r_1, i_1, 1) → Z
  45. Polygon(F, W, L, S) → yellow_1, y1A, y1D, y1B, y1C
  46. Polygon(S, L, Q, G) → green_1, g1B, g1C, g1D, g1A
  47. Polygon(L, U, A, Q) → red_1, r1B, r1C, r1D, r1A
  48. Polygon(U, L, W, C) → blue_1, b1D, b1A, b1B, b1C
  49. Dilate(D', equals / 100, A) → D''_1
  50. Vector(A, D''_1) → v_1
  51. Translate(L, v_1) → L'
  52. Translate(U, v_1) → U'
  53. Translate(A, v_1) → A'
  54. Translate(Q, v_1) → Q'
  55. Polygon(L', U', A', Q') → red_1', r1B', r1C', r1D', r1A'
  56. Dilate(P, equals / 100, W) → P'
  57. Vector(W, P') → j_1
  58. Translate(U, j_1) → U'_1
  59. Translate(L, j_1) → L'_1
  60. Translate(W, j_1) → W'
  61. Translate(C, j_1) → C'_1
  62. Polygon(U'_1, L'_1, W', C'_1) → blue_1', b1D', b1A', b1B', b1C'
  63. Dilate(K, equals / 100, J) → K'
  64. Vector(J, K') → m_1
  65. Translate(small_1, m_1) → small_1'
  66. Dilate(H, equals / 100, L) → H'
  67. Vector(L, H') → n_1
  68. Translate(F, n_1) → F'
  69. Translate(W, n_1) → W'_1
  70. Translate(L, n_1) → L'_2
  71. Translate(S, n_1) → S'
  72. Polygon(F', W'_1, L'_2, S') → yellow_1', y1A', y1D_1, y1B_1, y1C_1
  73. Dilate(B', equals / 100, G) → B''
  74. Vector(G, B'') → w
  75. Translate(S, w) → S'_1
  76. Translate(L, w) → L'_3
  77. Translate(Q, w) → Q'_1
  78. Translate(G, w) → G'
  79. Polygon(S'_1, L'_3, Q'_1, G') → green_1', g1B', g1C', g1D', g1A'
  80. Dilate(K, equals / 100, L) → K'_1
  81. Vector(L, K'_1) → u
  82. Translate(poly2, u) → poly2'
  83. Midpoint(A, H) → A_1
  84. Midpoint(I, H) → B_1
  85. Midpoint(B, I) → C_1
  86. Line(C_1, b) → d
  87. Line(P, a) → f_1
  88. Line(A_1, d) → l_1
  89. Line(B_1, f_1) → p_1
  90. Intersect(d, f_1) → D_1
  91. Intersect(l_1, p_1) → E_1
  92. Intersect(f_1, l_1) → F_1
  93. Intersect(d, p_1) → G_1
  94. Intersect(l_1, b_1) → H_1
  95. Circle(E, Segment[C_1, G_1]) → s_1
  96. Circle(D, Segment[C_1, G_1]) → t_1
  97. Circle(B, Segment[C_1, G_1]) → d_2
  98. Circle(C, Segment[C_1, G_1]) → e_2
  99. Intersect(b, e) → I_1
  100. Intersect(b, e) → J_1
  101. Intersect(b, e) → K_1
  102. Intersect(d_2, e, 1) → L_1
  103. Intersect(d_2, f, 1) → M_1
  104. Intersect(e_2, e, 1) → N_1
  105. Intersect(e_2, h, 1) → O_1
  106. Intersect(s_1, h, 1) → P_1
  107. Intersect(s_1, g, 1) → Q_1
  108. Intersect(t_1, g, 1) → R_1
  109. Intersect(t_1, f, 1) → S_1
  110. Polygon(D, S_1, J, Q_1) → yellow_2, y2B, y2C, y2D, y2A
  111. Polygon(S_1, J, L_1, B) → blue_2, b2A, b2D, b2C, b2B
  112. Polygon(L_1, J, O_1, C) → red_2, r2B, r2A, r2D, r2C
  113. Polygon(Q_1, E, O_1, J) → green_2, g2A, g2D, g2C, g2B
  114. Dilate(A, equals / 100, J) → A'_1
  115. Vector(J, A'_1) → u_1
  116. Dilate(B, equals / 100, J) → B'_1
  117. Vector(J, B'_1) → w_1
  118. Dilate(H, equals / 100, J) → H'_1
  119. Vector(J, H'_1) → u_2
  120. Dilate(I, equals / 100, J) → I'
  121. Vector(J, I') → v_2
  122. Translate(Q_1, v_2) → T_1
  123. Translate(E, v_2) → E'_1
  124. Translate(O_1, v_2) → U_1
  125. Translate(J, v_2) → J'
  126. Polygon(T_1, E'_1, U_1, J') → green_2', g2A', g2D', g2C', g2B'
  127. Translate(S_1, u_1) → V_1
  128. Translate(J, u_1) → J'_1
  129. Translate(L_1, u_1) → W_1
  130. Translate(B, u_1) → B'_2
  131. Polygon(V_1, J'_1, W_1, B'_2) → blue_2', b2A', b2D', b2C', b2B'
  132. Translate(L_1, w_1) → Z_1
  133. Translate(J, w_1) → J'_2
  134. Translate(O_1, w_1) → A_2
  135. Translate(C, w_1) → C'_2
  136. Polygon(Z_1, J'_2, A_2, C'_2) → red_2', z_1, r2A', r2D', r2C'
  137. Translate(D, u_2) → D'_1
  138. Translate(S_1, u_2) → B_2
  139. Translate(J, u_2) → J'_3
  140. Translate(Q_1, u_2) → C_2
  141. Polygon(D'_1, B_2, J'_3, C_2) → yellow_2', y2B', y2C', y2D', y2A'

AI Drawing Prompt Reference

Draw a right triangle ABC with AB as the hypotenuse and AC and BC as the legs. Use translation, rotation, and reflection to dissect and reassemble the triangle and related polygons to demonstrate that the sum of the squares of the legs equals the square of the hypotenuse. The steps include constructing the right triangle, creating polygons, translating vectors, rotating, and reflecting points, and ultimately forming four differently colored polygons corresponding to each part of the original triangle.
The prompt above was compiled from the original drawing commands and is for reference only.