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Proof Without Words: Pythagorean Theorem by Steve Phelps

Proof Without Words: Pythagorean Theorem by Steve Phelps
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2019-06-09 08:48
CirclePolygonAngleGeometric Construction
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This GeoGebra example provides a wordless proof of the Pythagorean Theorem, dynamically demonstrating the area relationships between squares on the sides of a right triangle. Using geometric constructions involving semicircles, polygons, segments, and intersection points, it visually confirms a² + b² = c². Key concepts include the Pythagorean Theorem, area computation, translation, and coordinate geometry. Construction employs GeoGebra commands such as Point, Semicircle, Segment, Polygon, Angle, Line, Intersect, OrthogonalLine, Circle, and conditional If commands. Ideal for middle to high school geometry classes, it helps students visualize and understand the theorem's proof through interactive dynamic geometry.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Semicircle(A, B) → c
  4. Point(c) → C
  5. Segment(A, C) → f
  6. Segment(C, B) → g
  7. Segment(B, A) → h
  8. Polygon(A, C, 4) → poly1, i, j, k, l, G, H
  9. Polygon(C, B, 4) → poly2, m, n, p, q, I, J
  10. Polygon(B, A, 4) → poly3, r, s, t, a, K, L
  11. Angle(A, C, B) → α
  12. Line(K, A) → b
  13. Line(L, B) → d
  14. Line(C, h) → e
  15. Intersect(d, p) → F
  16. Intersect(d, e) → M
  17. Intersect(b, e) → N
  18. Intersect(b, j) → D
  19. Intersect(e, k) → E
  20. Polygon(A, N, C) → t2, c_1, a_2, n_1
  21. Polygon(D, N, C) → t3, c_2, d_2, n_2
  22. Polygon(C, M, B) → t4, b_1, c_3, m_1
  23. Polygon(C, J, F, M) → q1, c_4, j_1, f_2, m_2
  24. Polygon(B, F, I) → t5, i_1, b_2, f_3
  25. OrthogonalLine(C, h) → k_1
  26. Intersect(k_1, t) → P
  27. Intersect(k_1, h) → Q
  28. Circle(H, Segment[G, D]) → d_1
  29. Intersect(d_1, l, 1) → O
  30. OrthogonalLine(O, h) → f_1
  31. Intersect(f_1, k) → R
  32. Polygon(O, R, G, D, A) → poly4, o, r_1, g_1, d_3, a_1
  33. Polygon(O, R, H) → t1, h_1, o_1, r_2
  34. If(0 < u ≤ 1, Polygon[Translate[O, Dilate[A, 1 - u, P] - A], Translate[R, Dilate[A, 1 - u, P] - A], Translate[G, Dilate[A, 1 - u, P] - A], Translate[D, Dilate[A, 1 - u, P] - A], Translate[A, Dilate[A, 1 - u, P] - A]], Polygon[Translate[O, P - A], Translate[R, P - A], Translate[G, P - A], Translate[D, P - A], Translate[A, P - A]]) → poly5
  35. If(2 < u ≤ 3, Polygon[Translate[A, A - Dilate[A, 3 - u, N]], Translate[N, A - Dilate[A, 3 - u, N]], Translate[C, A - Dilate[A, 3 - u, N]]], Polygon[Translate[A, A - N], Translate[N, A - N], Translate[C, A - N]]) → t6
  36. If(4 < u ≤ 5, Polygon[Translate[O, Q - Dilate[Q, 5 - u, R]], Translate[R, Q - Dilate[Q, 5 - u, R]], Translate[H, Q - Dilate[Q, 5 - u, R]]], Polygon[Translate[O, Q - R], Translate[R, Q - R], Translate[H, Q - R]]) → t7
  37. If(6 < u ≤ 7, Polygon[Translate[D, K - Dilate[K, 7 - u, N]], Translate[N, K - Dilate[K, 7 - u, N]], Translate[C, K - Dilate[K, 7 - u, N]]], Polygon[Translate[D, K - N], Translate[N, K - N], Translate[C, K - N]]) → t8
  38. If(8 < u ≤ 9, Polygon[Translate[B, P - Dilate[P, 9 - u, B]], Translate[F, P - Dilate[P, 9 - u, B]], Translate[I, P - Dilate[P, 9 - u, B]]], Polygon[Translate[B, P - B], Translate[F, P - B], Translate[I, P - B]]) → t9
  39. If(10 < u ≤ 11, Polygon[Translate[C, B - Dilate[B, 11 - u, M]], Translate[M, B - Dilate[B, 11 - u, M]], Translate[B, B - Dilate[B, 11 - u, M]]], Polygon[Translate[C, B - M], Translate[M, B - M], Translate[B, B - M]]) → t10
  40. If(12 < u ≤ 13, Polygon[Translate[C, L - Dilate[L, 13 - u, M]], Translate[J, L - Dilate[L, 13 - u, M]], Translate[F, L - Dilate[L, 13 - u, M]], Translate[M, L - Dilate[L, 13 - u, M]]], Polygon[Translate[C, L - M], Translate[J, L - M], Translate[F, L - M], Translate[M, L - M]]) → q2
  41. Segment(F, B) → l_1
  42. Segment(M, N) → p_1
  43. Segment(D, A) → q_1
  44. Segment(R, O) → s_1

AI Drawing Prompt Reference

Draw a right triangle and show squares on each of its sides. Include semicircles and line segments to represent area relationships. Use geometric constructions such as polygons, intersection points, angles, and perpendicular lines to demonstrate the proof of the Pythagorean theorem. Help students visualize the relationship a² + b² = c², making the theorem's proof dynamic and accessible.
The prompt above was compiled from the original drawing commands and is for reference only.