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Proof Without Words by Steve Phelps (16)

Proof Without Words by Steve Phelps (16)
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2015-02-15 09:25
TriangleCirclePolygonVectorGeometric Construction
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Explore the Pythagorean Theorem through dynamic geometry. Use points, circle arcs, polygons, and vectors in GeoGebra to observe changes and prove the theorem. Suitable for middle school and above students.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. CircleArc(B, D, A) → c
  4. Point(c) → E
  5. Polygon(A, E, 4) → poly1, a, b, d, e, F, G
  6. Polygon(E, C, 4) → poly2, f, g, h, i, H, I
  7. Polygon(C, A, 4) → poly3, j, k, l, m, J, K
  8. Polygon(A, E, F) → poly4, f_1, a_1, e_1
  9. Polygon(E, C, H) → poly5, h_1, e_2, c_1
  10. Polygon(C, A, J) → poly6, j_1, c_2, a_2
  11. OrthogonalLine(C, j_1) → n
  12. Intersect(n, c_1) → L
  13. Polygon(E, L, C) → poly7, c_3, e_3, l_1
  14. Midpoint(j_1) → M
  15. If(o ≤ 1, o, If[1 < o, 1, 0]) → r1
  16. Rotate(E, (-(((r1 * 180))°)), M) → E'
  17. Rotate(L, (-(((r1 * 180))°)), M) → L'
  18. Rotate(C, (-(((r1 * 180))°)), M) → C'
  19. Polygon(E', L', C') → poly7', c', e', l'
  20. Polygon(C, L, H) → poly8, h_2, c_4, l_2
  21. If(1 ≤ o ≤ 2, 2 - o, If[2 < o, 0, 1]) → d1
  22. Dilate(C, d1, A) → C'_1
  23. Vector(C, C'_1) → u
  24. Translate(C, u) → C'_2
  25. Translate(L, u) → L'_1
  26. Translate(H, u) → H'
  27. Polygon(C'_2, L'_1, H') → poly8', h', c'_2, l'_1
  28. If(1 ≤ o ≤ 2, o - 1, If[2 < o, 1, 0]) → r2
  29. Rotate(C'_2, (-(((r2 * 90))°)), C'_2) → N
  30. Rotate(L'_1, (-(((r2 * 90))°)), C'_2) → O
  31. Rotate(H', (-(((r2 * 90))°)), C'_2) → H''
  32. Polygon(N, O, H'') → poly8'', h'', n_1, o_1
  33. OrthogonalLine(E, n) → p
  34. Intersect(p, e_1) → P
  35. Polygon(P, A, E) → poly9, e_4, p_1, a_3
  36. If(2 ≤ o ≤ 3, 3 - o, If[3 < o, 0, 1]) → d2
  37. Dilate(E, d2, C) → E'_1
  38. Vector(E, E'_1) → v
  39. Translate(P, v) → P'
  40. Translate(A, v) → A'
  41. Translate(E, v) → E'_2
  42. Polygon(P', A', E'_2) → poly9', e'_2, p', a'
  43. Polygon(P, F, E) → poly10, e_5, p_2, f_2
  44. If(3 ≤ o ≤ 4, 4 - o, If[4 < o, 0, 1]) → d3
  45. Dilate(E, d3, J) → E'_3
  46. Vector(E, E'_3) → w
  47. Translate(P, w) → P'_1
  48. Translate(F, w) → F'
  49. Translate(E, w) → E'_4
  50. Polygon(P'_1, F', E'_4) → poly10', e'_4, p'_1, f'
  51. If(3 ≤ o ≤ 4, o - 3, If[4 < o, 1, 0]) → r3
  52. Rotate(P'_1, (-(((r3 * 90))°)), E'_3) → Q
  53. Rotate(F', (-(((r3 * 90))°)), E'_3) → F''
  54. Rotate(E'_4, (-(((r3 * 90))°)), E'_3) → R
  55. Polygon(Q, F'', R) → poly10'', r, q, f''
  56. Intersect(a_2, c') → S
  57. Polygon(S, C, H'') → poly11, h''_1, s, c_5
  58. If(4 ≤ o ≤ 5, o - 4, If[5 < o, 1, 0]) → r4
  59. Rotate(S, (-(((r4 * 180))°)), S) → S'
  60. Rotate(C, (-(((r4 * 180))°)), S) → C'_3
  61. Rotate(H'', (-(((r4 * 180))°)), S) → H'''
  62. Polygon(S', C'_3, H''') → poly11', h''', s', c'_3

AI Drawing Prompt Reference

Draw a right triangle and use arcs and squares to demonstrate the Pythagorean theorem. First, mark the points and intersection on the coordinate axes, then draw arcs and three squares on the sides of the triangle. By dynamically adjusting the positions and sizes of these shapes, observe their relationships and intuitively prove the theorem.
The prompt above was compiled from the original drawing commands and is for reference only.