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Classic Area Problems via the Converse of Pythagoras

Classic Area Problems via the Converse of Pythagoras
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2021-01-27 11:57
CirclePolygonAngleGeometric ConstructionLocus
This example demonstrates classic area-computation tasks by constructing geometric figures whose area relationships connect to the converse of the Pythagorean Theorem (and the right-angle structure behind it). The key mathematics includes: Pythagoras’ theorem and its converse, similarity of triangles (using proportional invariants to relate areas), and the way circle/semicircle configurations impose geometric constraints. Construction-wise, the GeoGebra setup uses Polygon, Segment, Line, and Ray to build the main boundaries; Intersect and Midpoint to determine key points; Angle to track angular relationships; OrthogonalLine to enforce perpendicular conditions tied to the Pythagorean configuration; Circle and Semicircle to create the circular area framework; Rotate to align or transform parts; and Point (and Locus when dynamic variation is needed) for interactive/animated behavior. It also employs dependent function objects (e.g., f, g, h, i) and UI controls (buttons/booleans) to switch views or parameter cases. Suitable for upper secondary geometry lessons, revision, and contest-level practice where “areas ↔ similarity ↔ right-angle criteria” is the learning goal.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Segment(P, B) → j
  3. Segment(P, C) → k
  4. Segment(P, A) → l
  5. OrthogonalLine(C, j) → m
  6. OrthogonalLine(A, j) → p
  7. OrthogonalLine(D, p) → q
  8. Line(P, B) → r
  9. Intersect(p, r) → E
  10. Intersect(m, r) → F
  11. Intersect(m, q) → G
  12. Intersect(p, q) → H
  13. Segment(B, F) → s
  14. Segment(A, E) → t
  15. Segment(D, H) → b
  16. Segment(C, G) → c
  17. Polygon(I, J, 4) → poly2, e, f_1, g_1, h_1, K, L
  18. Segment(M, J) → i_1
  19. Segment(M, I) → j_1
  20. Segment(M, K) → k_1
  21. Line(M, J) → l_1
  22. OrthogonalLine(I, l_1) → m_1
  23. OrthogonalLine(K, l_1) → n_1
  24. OrthogonalLine(L, m_1) → p_1
  25. Intersect(m_1, p_1) → N
  26. Intersect(l_1, m_1) → O
  27. Intersect(l_1, n_1) → Q
  28. Intersect(n_1, p_1) → R
  29. Segment(I, O) → q_1
  30. Segment(J, Q) → r_1
  31. Segment(K, R) → s_1
  32. Segment(L, N) → t_1
  33. Semicircle(A_3, B_3) → d
  34. Point(d) → T
  35. Segment(T, B_3) → a_1
  36. Ray(A_3, T) → b_1
  37. Point(a_1) → U
  38. Circle(A_3, Distance[U, B_3]) → c_1
  39. Intersect(c_1, b_1, 1) → D_3
  40. Circle(D_3, 10) → d_1
  41. Intersect(d_1, a_1, 1) → C_3
  42. Segment(D_3, C_3) → e_1
  43. Segment(C_3, B_3) → f_2
  44. Segment(A_3, B_3) → g_2
  45. Segment(A_3, D_3) → h_2
  46. Angle(A_3, T, B_3) → α
  47. Rotate(Translate[lpline3, Vector[A_3, B_3]], 90°, B_3) → lpline3_2
  48. Rotate(Translate[lpline3_2, Vector[A_3, B_3]], 90°, B_3) → lpline3_3
  49. Rotate(Translate[lpline3_3, Vector[A_3, B_3]], 90°, B_3) → lpline3_4
  50. Segment(A_4, B_4) → i_2
  51. Semicircle(A_4, B_4) → c_2
  52. Semicircle(B_4, A_4) → d_2
  53. Circle(A_4, 16) → e_2
  54. Intersect(c_2, e_2, 1) → C_4
  55. Segment(A_4, C_4) → j_2
  56. Segment(C_4, B_4) → k_2
  57. Semicircle(C_4, A_4) → p_2
  58. Semicircle(B_4, C_4) → q_2
  59. Midpoint(k_2) → O_1
  60. Midpoint(j_2) → O_2
  61. Intersect(p_2, i_2, 2) → D_4
  62. Point(larea41) → M_4
  63. Locus(M'_4, M_4) → loc1
  64. Point(larea42) → M4
  65. Locus(M4', M4) → loc2
  66. Point(larea43) → M3
  67. Locus(M3', M3) → loc3

AI Drawing Prompt Reference

Create a geometry diagram combining triangles with a semicircle and circles to support classic area relations. Start with a polygonal frame and draw three segments from a common point to selected polygon vertices. Construct several perpendicular lines from chosen vertices to one reference segment so that multiple intersection points form right-angle configurations. Then build a semicircle on the segment between two key points, pick a point on that semicircle using an intersection with a ray, and draw a circle centered at the semicircle point with a radius determined by the distance to another semicircle endpoint. Use the circle’s intersections with the ray to define additional segments. Mark the main angle at the key vertex and place two rotated copies of an auxiliary line to mirror the intended correspondence.
The prompt above was compiled from the original drawing commands and is for reference only.