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Proof Without Words: Dynamic Pythagorean Theorem Verification

Proof Without Words: Dynamic Pythagorean Theorem Verification
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2015-02-15 23:12
TriangleCircleGeometric Construction
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This example demonstrates the Pythagorean theorem through dynamic geometry. Key concepts include right triangles, circles, and geometric constructions. Construction involves GeoGebra commands like Point, Midpoint, Circle, and Intersect. Suitable for middle school students learning geometric proofs and interactive geometry.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Midpoint(A, B) → F
  4. Segment(F, B) → b
  5. Point(b) → C
  6. Semicircle(A, B) → c
  7. Semicircle(A, C) → d
  8. Point(d) → D
  9. CircumcircleSector(A, D, C) → e
  10. Point(c) → E
  11. CircumcircleSector(A, E, B) → f
  12. Point(xAxis) → G
  13. CircleSector(G, H, I) → h
  14. Semicircle(I, H) → k
  15. Segment(A, F) → i
  16. Segment(I, H) → j
  17. Circle(F, B) → p
  18. Circle(A, C) → q
  19. Intersect(p, q, 1) → J
  20. Midpoint(A, J) → K
  21. CircleSector(K, A, J) → r
  22. Midpoint(B, J) → L
  23. CircleSector(L, J, B) → s
  24. Midpoint(A, C) → M
  25. If(0 ≤ l ≤ 1, 1 - l, If[1 < l, 0, 1]) → d1
  26. Dilate(M, d1, K) → M'
  27. Vector(M, M') → u
  28. Translate(A, u) → A'
  29. Translate(D, u) → D'
  30. Translate(C, u) → C'
  31. CircumcircleSector(A', D', C') → e'
  32. Angle(B, A, J) → α
  33. If(0 ≤ l ≤ 1, l, If[1 < l, 1, 0]) → r1
  34. Rotate(A', ((180° - α) * r1), M') → A''
  35. Rotate(D', ((180° - α) * r1), M') → D''
  36. Rotate(C', ((180° - α) * r1), M') → C''
  37. CircumcircleSector(A'', D'', C'') → e''
  38. If(1 ≤ l ≤ 2, 2 - l, If[2 < l, 0, 1]) → d2
  39. Dilate(G, d2, L) → G'
  40. Vector(G, G') → v
  41. Translate(h, v) → h'
  42. Angle(A, B, J) → β
  43. If(1 ≤ l ≤ 2, l - 1, If[2 < l, 1, 0]) → r2
  44. Rotate(h', (-(((180° - β) * r2))), G') → h''
  45. Segment(A, B) → m

AI Drawing Prompt Reference

Draw a semicircle with a diameter on the x-axis. Pick a point on the semicircle and connect it to the diameter endpoints to form a right triangle. Using the right-angle vertex as center, draw two circles with radii equal to the two legs; they intersect at a point. Construct two circular sectors similar to the triangle at this intersection, then translate and rotate them along the legs to dynamically reassemble, visually showing that the sum of the areas of the squares on the legs equals the area of the square on the hypotenuse. The figure includes semicircles, circles, sectors, a triangle, and animated transformations.
The prompt above was compiled from the original drawing commands and is for reference only.