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Euclid's Pythagorean Proof

Euclid's Pythagorean Proof
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2017-11-30 02:17
TriangleVectorGeometric Construction
This example demonstrates Euclid's proof of the Pythagorean theorem. Key mathematical concepts include geometry, vectors, and polygons. Using GeoGebra commands such as Segment, Semicircle, and Polygon, an intuitive geometric proof is constructed. Suitable for middle school and above students to learn geometric proofs and the Pythagorean theorem.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Semicircle(A, B) → c
  3. Point(c) → C
  4. Segment(A, C) → b
  5. Segment(C, B) → d
  6. Polygon(A, C, 4) → poly1, e, f, g, h, D, E
  7. Polygon(C, B, 4) → poly2, i, j, k, l, F, G
  8. Polygon(B, A, 4) → poly3, m, n, p, q, H, I
  9. Segment(A, E) → r
  10. Segment(E, D) → s
  11. Segment(D, C) → t
  12. Segment(C, G) → a_1
  13. Segment(G, F) → b_1
  14. Segment(F, B) → c_1
  15. Segment(B, I) → d_1
  16. Segment(I, H) → e_1
  17. Segment(H, A) → f_1
  18. OrthogonalLine(C, a) → g_1
  19. Dilate(B, o, C) → B'
  20. Vector(C, B') → u
  21. Translate(D, u) → D'
  22. Polygon(E, D', B', A) → poly4, e_2, d', b', a_2
  23. Circle(B, Distance[C, D]) → h_1
  24. Rotate(A, (-(((90 * (o - 1)))°)), A) → A'
  25. Rotate(E, (-(((90 * (o - 1)))°)), A) → E'
  26. Rotate(B, (-(((90 * (o - 1)))°)), A) → B'_1
  27. Circle(C, Distance[A, B]) → k_1
  28. Intersect(k_1, g_1) → K, L
  29. Intersect(g_1, e_1) → M
  30. Dilate(M, o - 2, K) → M'
  31. Vector(K, M') → v
  32. Translate(C, v) → C'
  33. Polygon(A, C', M', H) → poly6, a_4, c', m', h_2
  34. Intersect(a, g_1) → N
  35. Dilate(A, o, C) → A'_1
  36. Vector(C, A'_1) → w
  37. Translate(G, w) → G'
  38. Polygon(F, G', A'_1, B) → poly7, f_2, g', a'_1, b_3
  39. Circle(A, Distance[C, G]) → p_1
  40. Rotate(A, ((90 * (o - 1)))°, B) → A'_2
  41. Rotate(F, ((90 * (o - 1)))°, B) → F'
  42. Rotate(B, ((90 * (o - 1)))°, B) → B'_2
  43. Polygon(B, C', M', I) → poly9, b_5, c'_1, m'_1, i_1
  44. Line(C, B) → l_1
  45. Line(A, C) → m_1
  46. Intersect(p_1, m_1) → J, O
  47. Intersect(h_1, l_1) → P, Q
  48. Polygon(E, P, B, A) → poly5, e_3, p_2, b_2, a_3
  49. Rotate(E, (-(((90 * (o - 1)))°)), A) → E'_1
  50. Rotate(P, (-(((90 * (o - 1)))°)), A) → P'
  51. Rotate(B, (-(((90 * (o - 1)))°)), A) → B'_3
  52. Rotate(A, (-(((90 * (o - 1)))°)), A) → A'_3
  53. Polygon(E'_1, P', B'_3, A'_3) → poly5', e'_1, p', b'_3, a'_3
  54. Polygon(A, O, F, B) → poly8, a_5, o_1, f_3, b_4
  55. Rotate(A, ((90 * (o - 1)))°, B) → A'_4
  56. Rotate(O, ((90 * (o - 1)))°, B) → O'
  57. Rotate(F, ((90 * (o - 1)))°, B) → F'_1
  58. Rotate(B, ((90 * (o - 1)))°, B) → B'_4
  59. Polygon(A'_4, O', F'_1, B'_4) → poly8', a'_4, o', f'_1, b'_4