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Pythagorean Theorem: A Visual Proof with Dynamic Geometry

Pythagorean Theorem: A Visual Proof with Dynamic Geometry
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2019-08-13 10:26
TrianglePolygonTransformationGeometric ConstructionSlider Animation
① This GeoGebra example demonstrates a visual proof of the Pythagorean theorem, showing that in a right triangle, the sum of the areas of the squares on the two legs equals the area of the square on the hypotenuse. ② Core mathematical concepts include the Pythagorean theorem, right triangles, square areas, rotation and translation transformations, as well as auxiliary elements like circumcircles and centroids. ③ Construction highlights: using GeoGebra commands such as Point, Ray, Circle, Segment, Intersect, Polygon, Rotate, Vector, and Translate, the applet builds multiple points, segments, circles, and polygons (e.g., R, A_1, B, f, G, c_2, D, E, etc.), then rotates and translates square pieces to visually rearrange them into equal areas. ④ Suitable for middle and high school mathematics classes (grades 8–12) for exploring the Pythagorean theorem, geometric transformations, and area-based proofs; it serves as an interactive teaching tool to deepen understanding of the theorem's geometric meaning.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A_1
  2. Point(yAxis) → B
  3. Ray(A_1, B) → f
  4. Point(f) → G
  5. Circle(A_1, G) → c_2
  6. Segment(A_1, D) → g
  7. Segment(G, E) → h
  8. Segment(E, D) → i
  9. Point(h) → Q
  10. Segment(G, Q) → j
  11. Circle(E, j) → d
  12. Circle(D, j) → e
  13. Circle(A_1, j) → k
  14. Intersect(d, i, 1) → H_1
  15. Intersect(e, g, 1) → I_1
  16. Intersect(k, f, 1) → J_1
  17. Polygon(Q, J_1, G) → t1, c, a, b
  18. Polygon(Q, H_1, E) → t2, e_1, f_2, h_1
  19. Polygon(H_1, I_1, D) → t3, d_1, h_2, i_1
  20. Polygon(J_1, I_1, A_1) → t4, a_1, j_2, i_2
  21. Rotate(Q, (-R), Q) → F'
  22. Rotate(H_1, (-R), Q) → H'
  23. Rotate(E, (-R), Q) → H
  24. Polygon(F', H', H) → t2', c_3, f', h'
  25. Rotate(H_1, R, I_1) → H'_1
  26. Rotate(I_1, R, I_1) → I'
  27. Rotate(D, R, I_1) → I_2
  28. Polygon(H'_1, I', I_2) → t3', d', h'_1, i'
  29. Point(Circle[G_1, j]) → K
  30. Segment(G_1, K) → l
  31. Point(l) → T
  32. Vector(G_1, T) → u
  33. Translate(H'_1, u) → M
  34. Translate(I', u) → I''
  35. Translate(I_2, u) → I
  36. Polygon(M, I'', I) → t3'', d'', m, i''
  37. Translate(J_1, u) → J'
  38. Translate(I_1, u) → I'_1
  39. Translate(A_1, u) → J
  40. Polygon(J', I'_1, J) → t4', a', j', i'_1
  41. Segment(G, A_1) → n