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Pythagorean Theorem Direct Proof

Pythagorean Theorem Direct Proof
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2016-10-05 04:40
TriangleVectorTrigonometryTransformationGeometric Construction
This example demonstrates a geometric proof of the Pythagorean Theorem through rotation and geometric transformations, illustrating the relationship between the sides of a right triangle. Core concepts include the Pythagorean Theorem, rotation, geometric transformations, and trigonometry. It uses commands like Segment, Polygon, Line, Intersect, Vector, Translate, Midpoint, Rotate in GeoGebra. Suitable for high school mathematics teaching and understanding geometric proofs.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Polygon(A, B, 4) → poly1, b, c, d, e, C, D
  3. Line(D, yAxis) → f
  4. Line(B, f) → g
  5. Line(A, xAxis) → h
  6. Line(C, h) → i
  7. Intersect(g, h) → E
  8. Polygon(A, B, E) → poly2, e_1, a_1, b_1
  9. Intersect(g, i) → F
  10. Polygon(B, C, F) → poly3, f_1, b_2, c_1
  11. Intersect(f, i) → G
  12. Polygon(C, D, G) → poly4, g_1, c_2, d_1
  13. Intersect(f, h) → H
  14. Polygon(D, A, H) → poly5, h_1, d_2, a_2
  15. Segment(I, J) → j
  16. Vector(I, J) → u
  17. Translate(G, u) → G'
  18. Vector(I, J) → v
  19. Translate(H, v) → H'
  20. Vector(I, J) → w
  21. Translate(F, w) → F'
  22. Vector(I, J) → k
  23. Translate(E, k) → E'
  24. Vector(I, J) → l
  25. Translate(C, l) → C'
  26. Translate(D, l) → D'
  27. Translate(G, l) → G'_1
  28. Polygon(C', D', G'_1) → poly4', g'_1, c', d'
  29. Vector(I, J) → m
  30. Translate(B, m) → B'
  31. Translate(C, m) → C'_1
  32. Translate(F, m) → F'_1
  33. Polygon(B', C'_1, F'_1) → poly3', f'_1, b', c'_1
  34. Midpoint(D', C') → K
  35. Midpoint(C', B') → L
  36. Rotate(C', 180°, K) → C''
  37. Rotate(D', 180°, K) → D''
  38. Rotate(G'_1, 180°, K) → M
  39. Polygon(C'', D'', M) → poly4'', m_1, c'', d''
  40. Rotate(B', 180°, L) → B''
  41. Rotate(C'_1, 180°, L) → N
  42. Rotate(F'_1, 180°, L) → O
  43. Polygon(B'', N, O) → poly3'', o, b'', n
  44. Rotate(D', 90°, M) → D''_1
  45. Polygon(H', D''_1, 4) → poly6, p, q, r, s, P, Q
  46. Polygon(D''_1, E', 4) → poly7, t, i_1, j_1, k_1, R, S