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Proof of the Pythagorean Theorem by Andrew Knauft

Proof of the Pythagorean Theorem by Andrew Knauft
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2016-01-10 17:15
TrianglePolygonTransformationGeometric Construction
This example demonstrates a visual proof of the Pythagorean theorem using dynamic geometry. It illustrates the relationship c² = a² + b² for a right triangle and verifies it via the algebraic identity c² - (b - a)² = 2ab. The construction employs GeoGebra commands Polygon, Rotate, and Segment to create the triangle and auxiliary figures, using objects a, b, c for length calculations. Designed for high school geometry and algebra classes, as well as junior‑level mathematics Olympiad students.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  2. Rotate(B, ((-3 * π / 2) * t), A) → B'_1
  3. Rotate(G, ((-3 * π / 2) * t), A) → G'
  4. Rotate(A, ((-3 * π / 2) * t), A) → A'
  5. Polygon(B'_1, G', A') → poly3', a', b'_1, g'
  6. Rotate(Polygon[Translate[B'_1, Vector[A', B'_1]], Translate[G', Vector[A', B'_1]], Translate[A', Vector[A', B'_1]]], π / 2 - (t * π), B'_1) → poly2
  7. Polygon(A, B, C, H, I) → poly3, e, f, g, h, i
  8. Segment(C, H) → j
  9. Segment(I, B) → k
  10. Polygon(G'_1, I, H, E) → poly4, g'_1, i_1, h_1, e_1

AI Drawing Prompt Reference

This is a visual proof of the Pythagorean theorem. The figure includes a right triangle and auxiliary figures to illustrate the relationship c² = a² + b². Key objects are the sides a, b, c of the right triangle and the auxiliary figures. The construction order is to first create the right triangle, then rotate and transform the auxiliary figures, and finally connect them to verify the relationship.
The prompt above was compiled from the original drawing commands and is for reference only.