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Phone Friendly: Pythagoras's Theorem

Phone Friendly: Pythagoras's Theorem
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2014-10-06 15:36
TriangleTransformationGeometric Construction
This example demonstrates the proof of Pythagoras's Theorem through shearing. It involves core mathematical concepts like Pythagorean Theorem, transformation, and vectors. The construction utilizes GeoGebra commands and objects such as Intersect, Point, Semicircle, Midpoint, Circle, Rotate, Line, Vector, Length, If, Polygon, and Angle. Suitable for high school mathematics teaching or self-study to understand the geometric proof of the theorem.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Semicircle(A, B) → h
  4. Midpoint(A, B) → O
  5. Circle(O, B) → o
  6. Point(h) → C
  7. Rotate(C, 90°, A) → D
  8. Rotate(A, 270°, C) → E
  9. Line(D, E) → m
  10. Vector(D, E) → v_b
  11. Length(v_b) → k_b
  12. Rotate(C, 270°, B) → F
  13. Rotate(B, 90°, C) → G
  14. Line(F, G) → n
  15. Intersect(m, n) → H
  16. Line(H, yAxis) → hc
  17. Vector(E, H) → u_a
  18. Length(u_a) → ku_b
  19. Vector(G, H) → u_b
  20. Length(u_b) → ku_a
  21. Vector(F, G) → v_a
  22. Length(v_a) → k_a
  23. Rotate(B, 270°, A) → I
  24. Vector(A, I) → v_c
  25. Length(v_c) → k_c
  26. Circle(H, (ku_a * ku_b / k_c)) → d
  27. Intersect(d, hc) → K, L
  28. Vector(H, L) → v_h
  29. Rotate(A, 90°, B) → J
  30. If(a < ka_1, Point[F, Vector[(a / ka_1 * u_b)]], If[a < ka_2, Point[F, u_b], Point[F, Vector[u_b + ((a - ka_2) / (1 - ka_2) * v_c)]]]) → F'
  31. Polygon(B, C, G, F) → P, b_1, c, g, f
  32. If(a < ka_1, Point[G, Vector[(a / ka_1 * u_b)]], If[a < ka_2, Point[H, Vector[((a - ka_1) / (ka_2 - ka_1) * v_h)]], Point[L, Vector[((a - ka_2) / (1 - ka_2) * v_c)]]]) → G'
  33. If(a < ka_1, C, If[a < ka_2, Point[C, Vector[((a - ka_1) / (ka_2 - ka_1) * v_h)]], Point[C, Vector[v_h + ((a - ka_2) / (1 - ka_2) * v_c)]]]) → Ca'
  34. Vector(G', Ca') → u
  35. If(a < ka_1, B, If[a < ka_2, B, Point[B, Vector[((a - ka_2) / (1 - ka_2) * v_c)]]]) → B'
  36. Vector(F', B') → v
  37. Polygon(B', Ca', G', F') → Q, b', c', g', f'
  38. If(b < kb_1, Point[E, Vector[(b / kb_1 * u_a)]], If[b < kb_2, Point[H, Vector[((b - kb_1) / (kb_2 - kb_1) * v_h)]], Point[L, Vector[((b - kb_2) / (1 - kb_2) * v_c)]]]) → E'
  39. If(b < kb_1, Point[D, Vector[(b / kb_1 * u_a)]], If[b < kb_2, Point[D, u_a], Point[D, Vector[u_a + ((b - kb_2) / (1 - kb_2) * v_c)]]]) → D'
  40. If(b < kb_1, C, If[b < kb_2, Point[C, Vector[((b - kb_1) / (kb_2 - kb_1) * v_h)]], Point[C, Vector[v_h + ((b - kb_2) / (1 - kb_2) * v_c)]]]) → Cb'
  41. If(b < kb_1, A, If[b < kb_2, A, Point[A, Vector[((b - kb_2) / (1 - kb_2) * v_c)]]]) → A'
  42. Polygon(A, C, E, D) → R, a_1, c_1, e, d_1
  43. Polygon(A', Cb', E', D') → S, a', cb', e', d'
  44. Polygon(A, B, J, I) → T, a_2, b_2, j, i
  45. Angle(A, C, B) → α
  46. Midpoint(C, B) → M
  47. Midpoint(A, C) → N
  48. Midpoint(a_2) → U

AI Drawing Prompt Reference

Draw a right triangle to demonstrate the Pythagorean theorem. Show the proof by shearing the triangle into three congruent smaller triangles. Label the lengths of each side and draw auxiliary lines, circles, and vectors. Include geometric elements like a semicircle, rotations, and vector length calculations to prove the relationship between the sides, clearly illustrating the theorem's geometric meaning.
The prompt above was compiled from the original drawing commands and is for reference only.