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Proof Without Words: Interactive Demonstration of the Pythagorean Theorem

Proof Without Words: Interactive Demonstration of the Pythagorean Theorem
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2015-02-06 03:52
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This GeoGebra applet provides a visual, drag‑and‑drop proof of the Pythagorean theorem without any textual explanation. By manipulating points and a slider, users can see how two right triangles are transformed and reassembled into a square using translation, dilation, rotation, and reflection, confirming a² + b² = c². The construction employs GeoGebra commands such as Point, Intersect, CircleArc, Polygon, Line, Segment, Vector, Translate, Dilate, Midpoint, Rotate, and Mirror, creating objects like A, B, C, D, poly1, etc. It is designed for high‑school geometry classes, teachers, and students preparing for math contests.

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, F) → poly1, f, a, e
  6. Polygon(E, G, C) → poly2, c_1, e_1, g
  7. Line(F, f) → b
  8. Line(G, g) → d
  9. Intersect(b, d) → H
  10. Line(H, E) → h
  11. Segment(A, C) → i
  12. Intersect(h, i) → I
  13. Vector(H, E) → u
  14. Translate(I, u) → I'
  15. Polygon(A, C, I') → poly3, i', a_1, c_2
  16. Dilate(F, If[0 ≤ n < 1, 1 - n, 0], H) → F'
  17. Dilate(G, If[0 ≤ n < 1, 1 - n, 0], H) → G'
  18. Polygon(A, F', E) → poly4, e_2, a_2, f'
  19. Polygon(E, G', C) → poly5, c_3, e_3, g'
  20. Dilate(E, If[1 ≤ n < 2, 2 - n, 0], I) → E'
  21. Vector(E, H) → v
  22. Translate(E', v) → E''
  23. Polygon(A, E'', E') → poly6, e', a_3, e''
  24. Polygon(E', E'', C) → poly7, c_4, e'_1, e''_1
  25. Midpoint(i') → J
  26. Rotate(A, (-((((n - 2) * 180))°)), J) → A'
  27. Rotate(E'', (-((((n - 2) * 180))°)), J) → E'''
  28. Rotate(E', (-((((n - 2) * 180))°)), J) → E''_1
  29. Polygon(A', E''', E''_1) → poly6', e''_2, a', e'''
  30. Rotate(E', (-((((n - 2) * 180))°)), J) → E''_2
  31. Rotate(E'', (-((((n - 2) * 180))°)), J) → E'''_1
  32. Rotate(C, (-((((n - 2) * 180))°)), J) → C'
  33. Polygon(E''_2, E'''_1, C') → poly7', c', e''_3, e'''_1
  34. Mirror(E'', B) → E'''_2
  35. Dilate(E'''_2, If[3 ≤ n < 4, 4 - n, 0], I') → K
  36. Mirror(I, B) → I'_1
  37. Polygon(I'_1, C, K) → poly8, k, i'_1, c_5
  38. Polygon(K, I'_1, A) → poly9, a_4, k_1, i'_2

AI Drawing Prompt Reference

Create an interactive visual proof of the Pythagorean theorem, showing two right triangles and a large square. Demonstrate how the right triangles transform and reassemble into the large square using translation, dilation, rotation, and reflection by manipulating points and a slider, visually confirming the a² + b² = c² relationship.
The prompt above was compiled from the original drawing commands and is for reference only.