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Proof of Pythagorean Theorem

Proof of Pythagorean Theorem
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2013-08-09 06:49
TriangleCircleGeometric Construction
In collections Pythagorean Theorem
This example demonstrates a geometric proof of the Pythagorean Theorem. The core math concepts involved are the relationship among the sides of a right triangle, expressed as a²+b²=c². The construction utilizes GeoGebra commands such as Segment, Point, Semicircle, Polygon, Vector, Rotate, Translate, OrthogonalLine, Intersect, and Angle. It is suitable for high school mathematics teaching or self-study to help understand the visual proof of the theorem.

GeoGebra Commands & Parameters

  1. Segment(F_1, G_1) → e
  2. Point(e) → B
  3. Semicircle(B, A) → c_1
  4. Point(c_1) → C
  5. Polygon(A, C, B) → poly1, b, c, α
  6. Vector(D, V) → u
  7. Rotate(A, 90*°, C) → A'
  8. Rotate(C, 90*°, C) → C'
  9. Rotate(B, 90*°, C) → B'
  10. Polygon(A', C', B') → poly1', b', a', c'
  11. Translate(A, u) → A'_1
  12. Translate(C, u) → C'_1
  13. Translate(B, u) → B'_1
  14. Polygon(A'_1, C'_1, B'_1) → poly1'_1, b'_1, a'_1, c'_1
  15. Vector(W, Z) → v
  16. Translate(A', v) → A''
  17. Translate(C', v) → C''
  18. Translate(B', v) → B''
  19. Polygon(A'', C'', B'') → poly1'', b'', a'', c''
  20. Vector(A_1, B_1) → w
  21. Rotate(A, 270*°, C) → A'_2
  22. Rotate(C, 270*°, C) → C'_2
  23. Rotate(B, 270*°, C) → B'_2
  24. Polygon(A'_2, C'_2, B'_2) → poly1'_2, b'_2, a'_2, c'_2
  25. Translate(A'_2, w) → C_1
  26. Translate(C'_2, w) → D_1
  27. Translate(B'_2, w) → E_1
  28. Polygon(C_1, D_1, E_1) → poly2, e_1, c_3, d_2
  29. Rotate(A, 180*°, C) → A'_3
  30. Rotate(C, 180*°, C) → C'_3
  31. Rotate(B, 180*°, C) → B'_3
  32. Polygon(A'_3, C'_3, B'_3) → poly1'_3, b'_3, a'_3, c'_3
  33. Vector(E, F) → z
  34. Translate(A'_3, z) → G
  35. Translate(C'_3, z) → H
  36. Translate(B'_3, z) → I
  37. Polygon(G, H, I) → poly3, i, g, h
  38. Polygon(D_1, H, C'', C'_1) → poly4, d_3, h_1, c''_1, c'_4
  39. Vector(J_1, K_1) → f
  40. Vector(O_1, P_1) → j
  41. Segment(F_1, B) → d
  42. Vector(J, K) → k
  43. Translate(D_1, k) → L
  44. Translate(H, k) → H'
  45. Translate(C'', k) → C'''
  46. Translate(C'_1, k) → M
  47. Polygon(L, H', C''', M) → poly4', l, h', c''', m
  48. Translate(A'', k) → A'''
  49. Translate(C'', k) → C'''_1
  50. Translate(B'', k) → B'''
  51. Polygon(A''', C'''_1, B''') → poly1''', b''', a''', c'''_1
  52. Translate(D_1, k) → O
  53. Translate(E_1, k) → P
  54. Translate(A'_1, f) → Q
  55. Translate(C'_1, f) → R
  56. Translate(B'_1, f) → S
  57. Polygon(Q, R, S) → poly5, s, q, r
  58. Translate(G, k) → G'
  59. Translate(H, k) → H'_1
  60. Translate(I, k) → I'
  61. Polygon(G', H'_1, I') → poly3', i', g', h'_1
  62. Translate(C_1, j) → T
  63. Translate(D_1, j) → U
  64. Translate(E_1, j) → H_1
  65. Polygon(T, U, H_1) → poly2'_1, h_2, t, u_1
  66. Polygon(Q_1, L_1, N_1, R_1) → poly6, q_1, l_1, n_1, r_1
  67. Polygon(I_1, M_1, S_1, N_1) → poly7, i_1, m_1, s_1, n_2
  68. OrthogonalLine(L, t) → a
  69. Intersect(t, a) → N
  70. Segment(L, N) → b_1
  71. Intersect(b_1, a''') → T_1
  72. Segment(T_1, N) → n
  73. Angle(B, C, A) → β

AI Drawing Prompt Reference

Draw a right triangle ABC with AB as the hypotenuse, then construct a semicircle with C as the center and AC as the radius. Next, construct a square that is tangent to the semicircle. Use translation and rotation operations to construct multiple similar figures to the original triangle, and use these figures to prove the Pythagorean theorem.
The prompt above was compiled from the original drawing commands and is for reference only.