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Visual Proof of Pythagoras' Theorem

Visual Proof of Pythagoras' Theorem
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2014-11-21 04:56
TriangleCircleCoordinate GeometryTransformationGeometric Construction
This example visually demonstrates the Pythagorean theorem through dynamic rotation and symmetry, providing an intuitive geometric proof. It involves core concepts like the Pythagorean theorem, rotation, symmetry, equations, and coordinate geometry. The construction utilizes commands and objects such as Rotate, Angle, Segment, Polygon, Midpoint, Line, Intersect, OrthogonalLine, Circle, and Point in GeoGebra. Suitable for high school mathematics instruction and exploratory learning.

GeoGebra Commands & Parameters

  1. Rotate(A, 90*°, B) → A'
  2. Angle(A, B, A') → α
  3. Segment(A, B) → a
  4. Segment(B, C) → b
  5. Segment(A, C) → c
  6. Polygon(A, B, 4) → poly1, a_1, b_1, d, e, D, E
  7. Polygon(B, C, 4) → poly2, b_2, c_1, f, g, F, G
  8. Polygon(C, A, 4) → poly3, c_2, a_2, h, i, H, I
  9. Segment(B, F) → j
  10. Midpoint(j) → J
  11. Line(J, c) → k
  12. Intersect(g, k) → K
  13. Intersect(k, c_1) → L
  14. Segment(K, L) → l
  15. OrthogonalLine(J, l) → m
  16. Intersect(m, f) → M
  17. Intersect(m, b) → N
  18. Intersect(f, b) → O
  19. Segment(M, N) → n
  20. Segment(A, I) → p
  21. Midpoint(p) → P
  22. Midpoint(D, A) → Q
  23. Segment(E, B) → q
  24. Circle(P, 5.1) → r
  25. Point(r) → R
  26. Point(r) → S
  27. Polygon(R, S, 4) → poly4, r_1, s, t, d_1, T, U
  28. Polygon(G, M, J, K) → poly5, g_1, m_1, j_1, k_1
  29. Line(R, U) → e_1
  30. Intersect(e_1, c) → V
  31. Line(R, S) → f_1
  32. Point(h) → W
  33. Line(S, W) → h_1
  34. Intersect(f_1, a_2) → Z
  35. Line(T, U) → i_1
  36. Intersect(i_1, h) → A_1
  37. Intersect(e_1, d_1) → B_1
  38. Intersect(i_1, i) → C_1
  39. Intersect(e_1, i) → D_1
  40. Intersect(c, c_2) → E_1
  41. Intersect(c, c_2) → F_1
  42. Intersect(c, c_2) → G_1
  43. Intersect(f_1, c) → H_1
  44. Polygon(K, J, N, B) → poly7, k_2, j_2, n_1, b_3
  45. Polygon(M, J, L, F) → poly6, m_2, j_3, l_1, f_2
  46. Polygon(J, N, C, L) → poly8, j_4, n_2, c_3, l_2
  47. Intersect(a_2, h_1) → I_1
  48. Polygon(V, C, C_1, U) → poly9, v, c_4, c_5, u
  49. Polygon(V, R, Z, A) → poly10, v_1, r_2, z, a_3
  50. Polygon(T, W, I, C_1) → poly11, t_1, w, i_2, c_6
  51. Polygon(S, Z, H, W) → poly12, s_1, z_1, h_2, w_1

AI Drawing Prompt Reference

Draw a right triangle ABC with A as the right angle vertex, B as the base vertex, and C as the hypotenuse vertex. Use rotation and symmetry to demonstrate the two legs a and b, and the hypotenuse c. Rotate point A 90 degrees around point B to get point A', construct angle α, and then use a series of polygons, segments, midpoints, intersections, etc., to finally show the area relationship of the right triangle, intuitively proving the Pythagorean theorem.
The prompt above was compiled from the original drawing commands and is for reference only.